Exponential and Logarithmic Relations Chapter Overview and Pacing

Transcription

1 Eponential and Logarithmic Relations Chapter verview and Pacing PACING (das) Regular Block Basic/ Basic/ Average Advanced Average Advanced Eponential Functions (pp. 5 50) Preview: Investigating Eponential Functions Graph eponential functions. Solve eponential equations and inequalities. LESSN BJECTIVES Logarithms and Logarithmic Functions (pp. 5 50) Evaluate arithmic epressions. (with 0- Solve arithmic equations and inequalities. Follow-Up) Follow-Up: Modeling Real-World Data: Curve Fitting Properties of Logarithms (pp. 5 56) Simplif and evaluate epressions using the properties of arithms. Solve arithmic equations using the properties of arithms. Common Logarithms (pp ) Solve eponential equations and inequalities using common arithms. Evaluate arithmic epressions using the Change of Base Formula. Follow-Up: Solving Eponential and Logarithmic Equations and Inequalities Base e and Natural Logarithms (pp ) Evaluate epressions involving the natural base and natural arithms. (with 0- (with 0- (with 0- (with 0- Solve eponential equations and inequalities using natural arithms. Follow-Up) Follow-Up) Follow-Up) Follow-Up) Eponential Growth and Deca (pp ) Use arithms to solve problems involving eponential deca. Use arithms to solve problems involving eponential growth. Stud Guide and Practice Test (pp ) Standardized Test Practice (pp ) Chapter Assessment TTAL Pacing suggestions for the entire ear can be found on pages T0 T. 50A Chapter 0 Eponential and Logarithmic Relations

2 Timesaving Tools Chapter Resource Manager All-In-ne Planner and Resource Center See pages T T. Stud Guide and Intervention CHAPTER 0 RESURCE MASTERS Practice (Skills and Average) Reading to Learn Mathematics Enrichment Assessment Applications* 5-Minute Check Transparencies Materials GCS (Preview: paper, scissors, grid paper, calculator) graphing calculator, grid paper, string Interactive Chalkboard AlgePASS: Tutorial Plus (lessons) SC posterboard (Follow-Up: graphing calculator, grid paper) , (Follow-Up: graphing calculator) SM plastic coins, paper currenc GCS 6, SC , *Ke to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual Chapter 0 Eponential and Logarithmic Relations 50B

3 Mathematical Connections and Background Continuit of Instruction Prior Knowledge Students have worked with eponents in man situations, including performing calculations, manipulating epressions, and appling properties. The have eplored properties of inverses for operations and for functions, and the have solved man kinds of equations and inequalities. This Chapter Students are introduced to the term arithm to solve for a variable that appears as an eponent. The eplore the relationship between eponents and arithms, and the use arithms with two special bases, base 0 or common arithms, and base e or natural arithms. The appl the Change of Base Formula to rewrite a arithm using a different base, and the appl appropriate formulas to solve problems involving eponential growth and eponential deca. Future Connections Students will continue to look at properties of and relationships between eponents and arithms. The will appl formulas for eponential growth and eponential deca in science courses and in consumer situations. The natural-base eponential function will have an important role in precalculus and calculus topics. Eponential Functions Eamine the list of characteristics for an eponential function on page 5. The first characteristic states that an eponential function is continuous and one-to-one. The term continuous means that the function can be traced without lifting our pencil. The term one-to-one means that a horizontal line passing through the graph will intersect no more than one point on the graph. This characteristic is important for the development of the arithmic function in Lesson 0-, since onl one-to-one functions can have inverses. The second characteristic listed is that the domain of the function is the set of all real numbers. This propert is important because it means that 5 has meaning, since 5 is a real number and part of the domain of. The third and fourth characteristics of an eponential function are related. The -ais is a horizontal asmptote of the graph of an eponential function. This means that the graph of this function approaches the horizontal line 0, getting closer and closer to this line but never crossing it. This restricts the graph of an eponential function to either Quadrants I and II, when a is positive, or to Quadrants III and IV, when a is negative. In terms of the range of the function, this means that when a is positive, all values of the function will be positive, and when a is negative, all values of the function will be negative. These two properties will also be important when considering the inverse of the eponential function. The last two properties are useful for graphing and writing eponential functions. Logarithms and Logarithmic Functions In the equation b, is referred to as the arithm, b is the base, and is sometimes referred to as the argument. The definition of a arithm given on page 5 indicates that a arithm is an eponent. When solving arithmic equations and inequalities, it is important to remember that a defining characteristic of a arithmic function is that its domain is the set of all positive numbers. This means that the arithm of 0 or of a negative number for an base is undefined. It is ver important to check possible solutions to arithmic equations in the original equation, to be sure that the would not result in taking the arithm of 0 or a negative number. For arithmic inequalities, this fact will eclude not just one value from the solution set, but a range of values. In Eample 8 on page 5, since the original inequalit asks for the values 0 ( ) and 0 ( 6), we must solve two inequalities, 0 and 6 0, to find what values must be ecluded from 50C Chapter 0 Eponential and Logarithmic Relations

4 the solution set we found using the Propert of Inequalit for Logarithmic Functions. Ecluding the values such that and 6, the solution set is all such that the following three inequalities are all satisfied:, 6, and 5. To simplif this compound inequalit, sketch all three inequalities, as shown below, and find where all three intersect The final number line shows that the solution set is the compound inequalit 5. Properties of Logarithms The word arithm is actuall a contraction of l o g ical a r i t h m etic. Logarithms were invented to make computation easier. Using arithms, multiplication changes to addition, according to the Product Propert of Logarithms, and division changes to subtraction, according to the Quotient Propert of Logarithms. This is illustrated in Eamples,, and of Lesson 0-. In these eamples, students are given the approimate value of specific arithms. Before the invention of the scientific calculator, these values took a good deal of time to compute. Rather than use the same arduous process to compute each and ever arithm one encountered, the properties of arithms allowed the use of a relative few arithmic values to compute others. Common Logarithms Before the invention of the scientific calculator, the appendices of algebra tets contained etensive tables of common arithms of numbers. In order to read these tables, ou had to understand the parts of a arithm. Ever arithm has two parts, the characteristic and the mantissa. A mantissa is the arithm of a number between and 0. When the original number is epressed in scientific notation, the characteristic is the power of 0. 65, Scientific notation 65,000 ( ) Propert of Equalit for Log Functions Product Propert Inverse Propert of Eponents and Logs Simplif. 65, characteristic mantissa Base e and Natural Logarithms Eponentiation, which is the inverse operation of taking a arithm, is sometimes referred to as finding the antiarithm. That is, if a then anti a. Since antiarithms mean the same operation as eponentiation, it follows that to find the antiarithm of a common arithm, ou would use nd [0 ] on a graphing calculator. To find the antiarithm of a natural arithm, antiln a, ou would use [e ]. nd Eponential Growth and Deca It is important to note that the variable r in the eponential deca formula a( r) t and the variable k in the alternate eponential deca formula ae kt are not equivalent. In a problem where a deca factor is given or asked for, the formula a( r) t should be used and not the formula ae kt. The same is true of the eponential growth formulas a( r) t and ae kt. Additional mathematical information and teaching notes are available in Glencoe s Algebra Ke Concepts: Mathematical Background and Teaching Notes, which is available at The lessons appropriate for this chapter are as follows. Eponential Functions (Lesson ) Growth and Deca (Lesson ) Chapter 0 Eponential and Logarithmic Relations 50D

5 and Assessment Tpe Student Edition Teacher Resources Techno/Internet ASSESSMENT INTERVENTIN ngoing Prerequisite Skills, pp. 5, 50, 58, 56, 55, 559 Practice Quiz, p. 58 Practice Quiz, p. 559 Mied Review Error Analsis Standardized Test Practice pen-ended Assessment Chapter Assessment pp. 5, 58, 56, 55, 559, 565 Find the Error, pp. 55, 5, 557 Common Misconceptions, p. 5 pp. 50, 57, 58, 56, 55, 559, 56, 56, 56, Writing in Math, pp. 50, 57, 56, 55, 559, 56 pen Ended, pp. 57, 55, 5, 59, 557, 56 Stud Guide, pp Practice Test, p Minute Check Transparencies Quizzes, CRM pp. 6 6 Mid-Chapter Test, CRM p. 65 Stud Guide and Intervention, CRM pp , , , 59 59, , Cumulative Review, CRM p. 66 Find the Error, TWE pp. 55, 5, 557 Unlocking Misconceptions, TWE pp. 5, 58 Tips for New Teachers, TWE p. 5 TWE p. 56 Standardized Test Practice, CRM pp Modeling: TWE pp. 50, 565 Speaking: TWE pp. 56, 559 Writing: TWE pp. 58, 55 pen-ended Assessment, CRM p. 6 Multiple-Choice Tests (Forms, A, B), CRM pp Free-Response Tests (Forms C, D, ), CRM pp Vocabular Test/Review, CRM p. 6 AlgePASS: Tutorial Plus Standardized Test Practice CD-RM standardized_test TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes vocabular_review Ke to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters Additional Intervention Resources The Princeton Review s Cracking the SAT & PSAT The Princeton Review s Cracking the ACT ALEKS TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment fleibilit: Worksheet Builder to make worksheet and tests Student Module to take tests on screen (optional) Management Sstem to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests. 50E Chapter 0 Eponential and Logarithmic Relations

6 Intervention Techno AlgePASS: Tutorial Plus CD-RM offers a complete, self-paced algebra curriculum. Algebra Lesson AlgePASS Lesson Eponential and Logarithmic Functions ALEKS is an online mathematics learning sstem that adapts assessment and tutoring to the student s needs. Subscribe at Intervention at Home Log on for student stud help. For each lesson in the Student Edition, there are Etra Eamples and Self-Check Quizzes. For chapter review, there is vocabular review, test practice, and standardized test practice For more information on Intervention and Assessment, see pp. T8 T. Reading and Writing in Mathematics Glencoe Algebra provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition Foldables Stud rganizer, p. 5 Concept Check questions require students to verbalize and write about what the have learned in the lesson. (pp. 57, 55, 5, 59, 557, 56, 566) Writing in Math questions in ever lesson, pp. 50, 57, 56, 55, 559, 56 WebQuest, pp. 59, 565 Teacher Wraparound Edition Foldables Stud rganizer, pp. 5, 566 Stud Notebook suggestions, pp. 5, 57, 55, 5, 59, 557, 56 Modeling activities, pp. 50, 565 Speaking activities, pp. 56, 559 Writing activities, pp. 58, 55 ELL Resources, pp. 50, 59, 57, 55, 550, 558, 56, 566 Additional Resources Vocabular Builder worksheets require students to define and give eamples for ke vocabular terms as the progress through the chapter. (Chapter 0 Resource Masters, pp. vii-viii) Reading to Learn Mathematics master for each lesson (Chapter 0 Resource Masters, pp. 577, 58, 589, 595, 60, 607) Vocabular PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabular lists that ou can customize. Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. Reading and Writing in the Mathematics Classroom WebQuest and Project Resources For more information on Reading and Writing in Mathematics, see pp. T6 T7. Chapter 0 Eponential and Logarithmic Relations 50F

7 Notes Have students read over the list of objectives and make a list of an words with which the are not familiar. Point out to students that this is onl one of man reasons wh each objective is important. thers are provided in the introduction to each lesson. Eponential and Logarithmic Relations Lessons 0- through 0- Simplif eponential and arithmic epressions. Lessons 0-, 0-, and 0-5 Solve eponential equations and inequalities. Lessons 0- and 0- Solve arithmic equations and inequalities. Lesson 0-6 Solve problems involving eponential growth and deca. Ke Vocabular eponential growth (p. 5) eponential deca (p. 5) arithm (p. 5) common arithm (p. 57) natural arithm (p. 55) Eponential functions are often used to model problems involving growth and deca. Logarithms can also be used to solve such problems. You will learn how a declining farm population can be modeled b an eponential function in Lesson 0-. NCTM Local Lesson Standards bjectives 0-,,, 6, 7, 8, Preview 0 0-,,,, 6, 8, 9, 0 0-,,,, 6, 7, 8, 9 0-,,, 5, 6, 8, Follow-Up 0 0-,,, 6, 7, 8, 9 0-,,, 6, 8, 9 0-,, Follow-Up 0-5,,,, 6, 7, 8, 9 0-6,,, 6, 8, 9 Ke to NCTM Standards: =Number & perations, =Algebra, =Geometr, =Measurement, 5=Data Analsis & Probabilit, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 0=Representation 50 Chapter 0 Eponential and Logarithmic Relations Vocabular Builder ELL The Ke Vocabular list introduces students to some of the main vocabular terms included in this chapter. For a more thorough vocabular list with pronunciations of new words, give students the Vocabular Builder worksheets found on pages vii and viii of the Chapter 0 Resource Masters. Encourage them to complete the definition of each term as the progress through the chapter. You ma suggest that the add these sheets to their stud notebooks for future reference when studing for the Chapter 0 test. 50 Chapter 0 Eponential and Logarithmic Relations

8 Prerequisite Skills To be successful in this chapter, ou ll need to master these skills and be able to appl them in problem-solving situations. Review these skills before beginning Chapter 0. Lessons 0- through 0- Multipl and Divide Monomials Simplif. Assume that no variable equals 0. (For review, see Lesson 5-.) (ab c ) 7a b c 6. 6 z z ab 7 z 5. 6 bc a 56 b c Lessons 0- and 0- Solve Inequalities Solve each inequalit. (For review, see Lesson -5) 5. a n a n 6 Lessons 0- and 0- Inverse Functions Find the inverse of each function. Then graph the function and its inverse. 9. f () (For review, see Lesson 7-8.) 9. See pp. 57A 57D for graphs. 9. f() 0. f(). f(). f() f () f () f () Lessons 0- and 0- Composition of Functions Find g[h()] and h[g()]. (For review, see Lesson 7-7.). h() g[h()]. h() 7 g[h()] 0 5 g() h[g()] g() 5 h[g()] h() g[h()] h() g[h()] 8 5 g() h[g()] g() h[g()] 8 This section provides a review of the basic concepts needed before beginning Chapter 0. Page references are included for additional student help. Prerequisite Skills in the Getting Read for the Net Lesson section at the end of each eercise set review a skill needed in the net lesson. For Prerequisite Lesson Skill 0- Composition of Functions (p. 50) 0- Multipling and Dividing Monomials (p. 58) 0- Solving Logarithmic Equations and Inequalities (p. 56) 0-5 Logarithmic Equations (p. 55) 0-6 Eponential Equations and Inequalities (p. 559) Make this Foldable to record information about eponential and arithmic relations. Begin with four sheets of grid paper. Fold and Cut First Sheets Second Sheets Fold and Label Insert first sheets through second sheets and align folds. Label pages with lesson numbers. Fold in half along the width. n the first two sheets, cut along the fold at the ends. n the second two sheets, cut in the center of the fold as shown. Reading and Writing As ou read and stud the chapter, fill the journal with notes, diagrams, and eamples for each lesson. Chapter 0 Eponential and Logarithmic Relations 5 For more information about Foldables, see Teaching Mathematics with Foldables. TM rganization of Data and Journal Writing After students make their Foldable journals, have them label two pages for each lesson in Chapter 0. Writers journals can be used b students to record the direction and progress of learning, to describe positive and negative eperiences during learning, to write about personal associations and eperiences called to mind during learning, and to list eamples of was in which new knowledge has or will be used in their dail life, as well as take notes, record ke concepts, and write eamples. Chapter 0 Eponential and Logarithmic Relations 5

9 Algebra Activit A Preview of Lesson 0- Getting Started bjective Use paper stacking to eplore an eponential function. Materials notebook paper scissors grid paper Teach You ma wish to do the eample as a demonstration while students complete the table on the chalkboard. Students ma recognize that the value is doubled for each successive cut, but the ma have to be led to realizing that this can be written in the form. Show students how to connect the points with a smooth curve, rather than connecting each pair of points with a straight line. Assess Have students work in small groups for Eercises 9. bserve students work to determine if the are able to write the function in Eercise 5. Students should conclude after Eercise 9 that eponential functions can increase faster than seems reasonable. Collect the Data Step Cut a sheet of notebook paper in half. Step Step Step Step 5 Stack the two halves, one on top of the other. Make a table like the one below and record the number of sheets of paper ou have in the stack after one cut. Number of Cuts Number of Sheets 0 A Preview of Lesson 0- Investigating Eponential Functions Cut the two stacked sheets in half, placing the resulting pieces in a single stack. Record the number of sheets of paper in the new stack after cuts. Continue cutting the stack in half, each time putting the resulting piles in a single stack and recording the number of sheets in the stack. Stop when the resulting stack is too thick to cut. Analze the Data. Write a list of ordered pairs (, ), where is the number of cuts and is the number of sheets in the stack. Notice that the list starts with the ordered pair (0, ), which represents the single sheet of paper before an cuts were made.. Continue the list, beond the point where ou stopped cutting, until ou reach the ordered pair for 7 cuts. Eplain how ou calculated the last values for our list, after ou had stopped cutting.. Plot the ordered pairs in our list on a coordinate grid. Be sure to choose a scale for the -ais so that ou can plot all of the points. See pp. 57A 57D. cuts.. Describe the pattern of the points ou have plotted. Do the lie on a straight line? The points do not lie in a straight line. The slope increases as the values increase. Make a Conjecture 5. Write a function that epresses as a function of. 6. Use a calculator to evaluate the function ou wrote in Eercise 5 for 8 and 9. Does it give the correct number of sheets in the stack after 8 and 9 cuts? 7. Notebook paper usuall stacks about 500 sheets to the inch. How thick would our stack of paper be if ou had been able to make 9 cuts? about in. 8. Suppose each cut takes about 5 seconds. If ou had been able to keep cutting,. (0, ), (, ), (, ), (, 8), (, 6),. (5, ), (6, 6), (7, 8); The value is found b raising to the number of 56, 5; es ou would have made 6 cuts in three minutes. At 500 sheets to the inch, make a conjecture as to how thick ou think the stack would be after 6 cuts. Sample answer: 9. Use our function from Eercise 5 to calculate the thickness of our stack after million ft 6 cuts. Write our answer in miles. 69 mi Stud Notebook You ma wish to have students summarize this activit and what the learned from it. 5 Investigating Slope-Intercept Form 5 Chapter 0 Eponential and Logarithmic Relations Resource Manager Teaching Algebra with Manipulatives p. (grid paper) p. 75 (student recording sheet) Glencoe Mathematics Classroom Manipulative Kit scissors coordinate grid stamp 5 Chapter 0 Eponential and Logarithmic Relations

10 Eponential Functions Lesson Notes Vocabular eponential function eponential growth eponential deca eponential equation eponential inequalit Stud Tip Common Misconception Be sure not to confuse polnomial functions and eponential functions. While and each have an eponent, is a polnomial function and is an eponential function. Graph eponential functions. Solve eponential equations and inequalities. EXPNENTIAL FUNCTINS In an eponential function like, the base is a constant, and the eponent is a variable. Let s eamine the graph of. Eample does an eponential function describe tournament pla? The NCAA women s basketball tournament begins with 6 teams and consists of 6 rounds of pla. The winners of the first round pla against each other in the second round. The winners then move from the Sweet Siteen to the Elite Eight to the Final Four and finall to the Championship Game. The number of teams that compete in a tournament of rounds is. Graph an Eponential Function Sketch the graph of. Then state the function s domain and range. Make a table of values. Connect the points to sketch a smooth curve SW Mo. State WEST Washington Xavier MIDEAST Purdue As the value of decreases, the value of approaches 0. 8 The domain is all real numbers, while the range is all positive numbers. 00 NCAA Women s Tournament 7 6. SW Missouri State Purdue Purdue Notre Dame Notre Dame Connecticut Notre Dame 7 Notice that the domain of includes irrational numbers such as 7. Lesson 0- Eponential Functions 5 Connecticut EAST La. Tech Notre Dame MIDWEST Vanderbilt Focus 5-Minute Check Transparenc 0- Use as a quiz or review of Chapter 9. Mathematical Background notes are available for this lesson on p. 50C. Building on Prior Knowledge Ask students where the have heard the term eponential before and what the think it might mean. Students ma have heard terms like eponential growth on a television news program and the might think that eponential means enormous. Use students answers to introduce the concept of eponential functions. does an eponential function describe tournament pla? Ask students: How man winners are there in the first round of the tournament? After each round, how has the number of teams changed? The number of teams remaining after each round is half the number of teams that plaed in that round. If the tournament field was reduced to teams, how man basketball games would have to be plaed b the tournament s winning team? 5 games Resource Manager Workbook and Reproducible Masters Chapter 0 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 575 Practice, p. 576 Reading to Learn Mathematics, p. 577 Enrichment, p. 578 Graphing Calculator and Spreadsheet Masters, p. 5 Teaching Algebra With Manipulatives Masters, pp Transparencies 5-Minute Check Transparenc 0- Answer Ke Transparencies Techno Interactive Chalkboard Lesson - Lesson Title 5

11 Teach EXPNENTIAL FUNCTINS In-Class Eample Power Point Teaching Tip Watch for students who do not understand wh the graph in Eample cannot simpl be modeled b a quadratic, cubic, or quartic function. Point out that the graphs of,, and all pass through the point (0, 0) and not through the point (0, ). Sketch the graph of. Then state the function s domain and range.. See pp. 57A 57D. You can use a TI-8 Plus graphing calculator to look at the graph of two other eponential functions, and. Families of Eponential Functions The calculator screen shows the graphs of and. Think and Discuss,5. See margin. ( ). How do the shapes of the graphs compare?. How do the asmptotes and -intercepts of the graphs compare? [5, 5] scl: b [, 8] scl:. Describe the relationship between the graphs.. Graph each group of functions on the same screen. Then compare the graphs, listing both similarities and differences in shape, asmptotes, domain, range, and -intercepts. a.,, and b.,, and c. () and () ; () and. 5. Describe the relationship between the graphs of () and. The domain is all real numbers, while the range is all positive numbers. Answers. The shapes of the graphs are the same.. The asmptote for each graph is the -ais and the -intercept for each graph is.. The graphs are reflections of each other over the -ais. 5. The graphs are reflections of each other over the -ais. Stud Tip Look Back To review continuous functions, see page 6, Eercises 60 and 6. To review one-to-one functions, see Lesson -. Stud Tip Eponential Growth and Deca Notice that the graph of an eponential growth function rises from left to right. The graph of an eponential deca function falls from left to right. In general, an equation of the form ab, where a 0, b 0, and b, is called an eponential function with base b. Eponential functions have the following characteristics.. The function is continuous and one-to-one.. The domain is the set of all real numbers.. The -ais is an asmptote of the graph.. The range is the set of all positive numbers if a 0 and all negative numbers if a The graph contains the point (0, a). That is, the -intercept is a. 6. The graphs of ab and a b are reflections across the -ais. There are two tpes of eponential functions: eponential growth and eponential deca. The base of an eponential growth function is a number greater than one. The base of an eponential deca function is a number between 0 and. Eponential Deca Eponential Growth and Deca If a 0 and b, the function ab represents eponential growth. If a 0 and 0 b, the function ab represents eponential deca. Eponential Growth 5 Chapter 0 Eponential and Logarithmic Relations Families of Eponential Functions Have students begin b graphing the two functions separatel, so the recognize that the two curves shown in the book are two distinct graphs. Students are used to seeing the U-shaped graphs of polnomial functions and might have difficult separating the graphs visuall. Also, a reminder about the meaning of the term asmptotes ma be helpful for man students. 5 Chapter 0 Eponential and Logarithmic Relations

12 Farming In 999, 7% of the net farm income in the United States was from direct government paments. The USDA has set a goal of reducing this percent to % b 005. Source: USDA TEACHING TIP In Eample, one of the given points is the -intercept. You ma wish to give our students a challenge problem in which an two points are given and students use a sstem of equations to find the equation of the eponential function. Eample Identif Eponential Growth and Deca Determine whether each function represents eponential growth or deca. Function Eponential functions are frequentl used to model the growth or deca of a population. You can use the -intercept and one other point on the graph to write the equation of an eponential function. Eample Eponential Growth or Deca? a. 5 The function represents eponential deca, since the base,, is between 0 and. 5 b. () The function represents eponential growth, since the base,, is greater than. c. 7(.) The function represents eponential growth, since the base,., is greater than. Write an Eponential Function FARMING In 98, there were 0,000 farms in Minnesota, but b 998, this number had dropped to 80,000. a. Write an eponential function of the form ab that could be used to model the farm population of Minnesota. Write the function in terms of, the number of ears since 98. For 98, the time equals 0, and the initial population is 0,000. Thus, the -intercept, and value of a, is 0,000. For 998, the time equals or 5, and the population is 80,000. Substitute these values and the value of a into an eponential function to approimate the value of b. ab Eponential function 80,000 0,000b 5 Replace with 5, with 80,000, and a with 0, b 5 Divide each side b 0, b Take the 5th root of each side. To find the 5th root of 0.78, use selection 5: under the MATH menu on the TI-8 Plus. KEYSTRKES: ENTER An equation that models the farm population of Minnesota from 98 to 998 is 0,000(0.98). b. Suppose the number of farms in Minnesota continues to decline at the same rate. Estimate the number of farms in 00. For 00, the time equals or 7. 0,000(0.98) Modeling equation 0,000(0.98) 7 Replace with 7. 59,5 Use a calculator. The farm population in Minnesota will be about 59,5 in 00. Teacher to Teacher David S. Daniels Lesson 0- Eponential Functions 55 Longmeadow H.S., Longmeadow, MA As a lead-in activit for eponential functions, have students flip 50 pennies and count the number of heads. Then have students remove those pennies that landed on heads and repeat the activit. Students should record their results and make a plot of the trial number versus the number of heads counted in that trial. The graph will model that of ( ). In-Class Eamples Power Point Determine whether each function represents eponential growth or deca. a. (0.7) The function represents eponential deca, since the base, 0.7, is between 0 and. b. () The function represents eponential growth, since the base,, is greater than. c. 0 The function represents eponential growth, since the base,, is greater than. CELLULAR PHNES In December of 990, there were 5,8,000 cellular telephone subscribers in the United States. B December of 000, this number had risen to 09,78, 000. Source: Cellular Telecommunications Industr Association a. Write an eponential function of the form ab that could be used to model the number of cellular telephone subscribers in the U.S. Write the function in terms of, the number of ears since ,8,000(.5) b. Suppose the number of cellular telephone subscribers continues to increase at the same rate. Estimate the number of U.S. subscribers in 00. about,6,000,000 subscribers Interactive Chalkboard PowerPoint Presentations This CD-RM is a customizable Microsoft PowerPoint presentation that includes: Step-b-step, dnamic solutions of each In-Class Eample from the Teacher Wraparound Edition Additional, Your Turn eercises for each eample The 5-Minute Check Transparencies Hot links to Glencoe nline Stud Tools Lesson 0- Eponential Functions 55

13 EXPNENTIAL EQUATINS AND INEQUALITIES In-Class Eamples Simplif each epression. a b. (6 5 ) Solve each equation. a. 9n 56 n b. 5 9 Power Point Stud Tip Look Back To review Properties of Power, see Lesson 5-. EXPNENTIAL EQUATINS AND INEQUALITIES Since the domain of an eponential function includes irrational numbers such as, all the properties of rational eponents appl to irrational eponents. Eample Simplif each epression. a. 5 Simplif Epressions with Irrational Eponents 5 5 Product of Powers b Smbols 7 6 Power of a Power Product of Radicals The following propert is useful for solving eponential equations. equations are equations in which variables occur as eponents. Propert of Equalit for Eponential Functions If b is a positive number other than, then b b if and onl if. Eample If 8, then 8. Eample 5 Solve Eponential Equations Solve each equation. a. n 8 n 8 riginal equation n Rewrite 8 as so each side has the same base. n Propert of Equalit for Eponential Functions n Subtract from each side. n Divide each side b. The solution is. CHECK n 8 riginal equation b. 8 8 Substitute for n. 8 Simplif. 8 8 Simplif. 8 riginal equation ( ) ( ) Rewrite each side with a base of. ( ) Power of a Power ( ) Propert of Equalit for Eponential Functions Distributive Propert Subtract from each side. The solution is. Eponential 56 Chapter 0 Eponential and Logarithmic Relations 56 Chapter 0 Eponential and Logarithmic Relations

14 Concept Check a. quadratic b. eponential c. linear d. eponential The following propert is useful for solving inequalities involving eponential functions or eponential inequalities. Smbols Propert of Inequalit for Eponential Functions If b, then b b if and onl if, and b b if and onl if. Eample If 5 5, then. Eample 6 This propert also holds for and. Solve Eponential Inequalities Solve p 56. p riginal inequalit 56 p Rewrite 56 as or so each side has the same base. p p Propert of Inequalit for Eponential Functions Add to each side. p Divide each side b. The solution set is p. CHECK Test a value of p greater than ; for eample, p 0. p 56 riginal inequalit (0)? 56 Replace p with 0.? 56 Simplif. 56 a a. PEN ENDED Give an eample of a value of b for which b represents eponential deca. Sample answer: 0.8. Identif each function as linear, quadratic, or eponential. a. b. () c. d. (0.) Match each function with its graph.. 5 c. (5) a 5. 5 b a. b. c. In-Class Eample 6 Practice/Appl Stud Notebook Power Point Solve 5 k 6 5. k 7 Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 0. include eamples of eponential growth and deca graphs and equations. include an other item(s) that the find helpful in mastering the skills in this lesson. Answers 6. D { is all real numbers.}, R { 0} () Guided Practice 6 7. See margin. Sketch the graph of each function. Then state the function s domain and range. 6. () 7. Lesson 0- Eponential Functions D { is all real numbers.}, R { 0} ( ) Differentiated Instruction Auditor/Musical Going around the room, have students count b ones beginning at, with each student calling out one number. Instruct them to record the number the called as n. Then have students find n and n. Now go around the room again and ask students to state their value of n (for a class of 0 students, the recited numbers are all the squares from to 96). Now have students state their values of n (for a class of 0, the recited numbers are all the powers of from to or about 0 9 ). Lesson 0- Eponential Functions 57

15 About the Eercises rganization b bjective Eponential Functions: 8, 57 6 Eponential Equations and Inequalities: 7 56, 6 66 dd/even Assignments Eercises 56 are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercise 6 involves research on the Internet or other reference materials. Eercises 7 75 require the use of graphing calculators. Assignment Guide Basic:,, 7 5 odd, 57 6, 68 70, Average: 55 odd, 59 6, 67 70, (optional: 7 75) Advanced: 56 even, 6 86 (optional: 87 89) GUIDED PRACTICE KEY Eercises Eamples 6, 7 8 0,, 9, , 6 Application indicates increased difficult Practice and Appl Homework Help For Eercises See Eamples 6 7 8, , 6 Etra Practice See page 89. Determine whether each function represents eponential growth or deca. 8. (7) growth 9. (0.5) deca 0. 0.(5) growth Write an eponential function whose graph passes through the given points.. (0, ) and (, 6). (0, 8) and (, ) 8() Simplif each epression or 7. (a ) a 5. 8 or 7 Solve each equation or inequalit. Check our solution. 6. n ANIMAL CNTRL For Eercises 9 and 0, use the following information. During the 9th centur, rabbits were brought to Australia. Since the rabbits had no natural enemies on that continent, their population increased rapidl. Suppose there were 65,000 rabbits in Australia in 865 and,500,000 in Write an eponential function that could be used to model the rabbit population in Australia. Write the function in terms of, the number of ears since ,000(6.0) 0. Assume that the rabbit population continued to grow at that rate. Estimate the Australian rabbit population in 87.,890,95,000 Sketch the graph of each function. Then state the function s domain and range. 6. See pp. 57A 57D.. (). 5(). 0.5() (5) Determine whether each function represents eponential growth or deca. 7. 0(.5) growth 8. () growth deca 0. 5 growth. 0 deca. 0.(5) deca 58 Chapter 0 Eponential and Logarithmic Relations Write an eponential function whose graph passes through the given points.. (0, ) and (, ). (0, ) and (, 5) (5) 5. (0, 7) and (, 6) 7() 6. (0, 5) and (, 5) 5 7. (0, 0.) and (, 5.) 0.() 8. (0, 0.) and (5, 9.6) 0.() Simplif each epression or n n n. 6 5 Solve each equation or inequalit. Check our solution. 5. p 5. n n 5 8. n n m 8 m n 8 n n p 6 p 5. 5p 6 5p , 5 Answers (p. 59) million; 7.6 million;. million; These answers are in close agreement with the actual populations in those ears million; 8. million; No, the growth rate has slowed considerabl. The population in 000 was much smaller than the equation predicts it would be. 58 Chapter 0 Eponential and Logarithmic Relations

16 The magnitude of an earthquake can be represented b an eponential equation. Visit com/webquest to continue work on our WebQuest project. 6. Eponential; the r base,, is n fied, but the eponent, nt, is variable since the time t can var. BILGY For Eercises 57 and 58, use the following information. The number of bacteria in a colon is growing eponentiall. 57. Write an eponential function to model the population of bacteria hours after P.M. 00(6.) 58. How man bacteria were there at 7 P.M. that da? about,008,90 Log Time Number of Bacteria P.M. 00 P.M. 000 PPULATIN For Eercises 59 6, use the following information. Ever ten ears, the Bureau of the Census counts the number of people living in the United States. In 790, the population of the U.S. was.9 million. B 800, this number had grown to 5. million. 59. Write an eponential function that could be used to model the U.S. population in millions for 790 to 800. Write the equation in terms of, the number of decades since (.5) 60. Assume that the U.S. population continued to grow at that rate. Estimate the population for the ears 80, 80, and 860. Then compare our estimates with the actual population for those ears, which were 9.6, 7.06, and. million, respectivel. See margin. 6. RESEARCH Estimate the population of the U.S. in 000. Then use the Internet or other reference to find the actual population of the U.S. in 000. Has the population of the U.S. continued to grow at the same rate at which it was growing in the earl 800s? Eplain. See margin. Stud Guide and Intervention, p. 57 (shown) and p. 57 NAME DATE PERID 0- Stud Guide and Intervention Eponential Functions Eponential Functions An eponential function has the form ab, where a 0, b 0, and b. Properties of an Eponential Function Eponential Growth and Deca. The function is continuous and one-to-one.. The domain is the set of all real numbers.. The -ais is the asmptote of the graph.. The range is the set of all positive numbers if a 0 and all negative numbers if a The graph contains the point (0, a). If a 0 and b, the function ab represents eponential growth. If a 0 and 0 b, the function ab represents eponential deca. Eample Sketch the graph of 0.(). Then state the function s domain and range. Make a table of values. Connect the points to form a smooth curve The domain of the function is all real numbers, while the range is the set of all positive real numbers. Eample Determine whether each function represents eponential growth or deca. a. 0.5() b..8() c..(0.5) eponential growth, neither, since.8, eponential deca, since since the base,, is the value of a is less the base, 0.5, is between greater than than 0. 0 and Eercises Sketch the graph of each function. Then state the function s domain and range.. ().. 0.5(5) Domain: all real Domain: all real Domain: all real numbers; Range: all numbers; Range: all numbers; Range: all positive real numbers negative real numbers positive real numbers Determine whether each function represents eponential growth or deca.. 0.(.) growth 5. 5 neither 6. (0) 5 deca 0- NAME 57 DATE PERID Practice (Average) Skills Practice, p. 575 and Practice, p. 576 (shown) Eponential Functions Sketch the graph of each function. Then state the function s domain and range...5(). (). (0.5) Lesson 0- Computers Since computers were invented, computational speed has multiplied b a factor of about ever three ears. Source: MNEY For Eercises 6 6, use the following information. Suppose ou deposit a principal amount of P dollars in a bank account that pas compound interest. If the annual interest rate is r (epressed as a decimal) and the bank makes interest paments n times ever ear, the amount of mone A ou would have after t ears is given b A(t) P n r nt. 6. If the principal, interest rate, and number of interest paments are known, r nt what tpe of function is A(t) P n? Eplain our reasoning. 6. Write an equation giving the amount of mone ou would have after t ears if ou deposit $000 into an account paing % annual interest compounded quarterl (four times per ear). A(t) 000(.0) t 6. Find the account balance after 0 ears. $6.7 CMPUTERS For Eercises 65 and 66, use the information at the left. 65. If a tpical computer operates with a computational speed s toda, write an epression for the speed at which ou can epect an equivalent computer to operate after three-ear periods. s 66. Suppose our computer operates with a processor speed of 600 megahertz and ou want a computer that can operate at 800 megahertz. If a computer with that speed is currentl unavailable for home use, how long can ou epect to wait until ou can bu such a computer?.5 three-ear periods or.5 r domain: all real domain: all real domain: all real numbers; range: all numbers; range: all numbers; range: all positive numbers positive numbers positive numbers Determine whether each function represents eponential growth or deca.. 5(0.6) deca 5. 0.() growth 6. 5 deca Write an eponential function whose graph passes through the given points. 7. (0, ) and (, ) 8. (0, ) and (, 0) 9. (0, ) and (,.5) (5) (0.5) 0. (0, 0.8) and (,.6). (0, 0.) and (, 0). (0, ) and (, 8) 0.8() 0.(5) () Simplif each epression.. ( ) 8 6. (n ) 75 n n n n Solve each equation or inequalit. Check our solution n 5 9 n n n n n. 6 n 8n BILGY For Eercises 5 and 6, use the following information. The initial number of bacteria in a culture is,000. The number after das is 96, Write an eponential function to model the population of bacteria after das.,000() 6. How man bacteria are there after 6 das? 768, EDUCATIN A college with a graduating class of 000 students in the ear 00 predicts that it will have a graduating class of 86 in ears. Write an eponential function to model the number of students in the graduating class t ears after (.05) t Reading to Learn Mathematics, p NAME 576 DATE PERID Reading to Learn Mathematics Eponential Functions Pre-Activit How does an eponential function describe tournament pla? Read the introduction to Lesson 0- at the top of page 5 in our tetbook. How man rounds of pla would be needed for a tournament with 00 plaers? 7 ELL 67. Sometimes; true when b, but false when b CRITICAL THINKING Decide whether the following statement is sometimes, alwas, or never true. Eplain our reasoning. For a positive base b other than, b b if and onl if. Lesson 0- Eponential Functions 59 Reading the Lesson. Indicate whether each of the following statements about the eponential function 0 is true or false. a. The domain is the set of all positive real numbers. false b. The -intercept is. true c. The function is one-to-one. true d. The -ais is an asmptote of the graph. false e. The range is the set of all real numbers. false Lesson 0- NAME DATE PERID 0- Enrichment, p. 578 Finding Solutions of Perhaps ou have noticed that if and are interchanged in equations such as and, the resulting equation is equivalent to the original equation. The same is true of the equation. However, finding solutions of and drawing its graph is not a simple process. Solve each problem. Assume that and are positive real numbers.. If a 0, will (a, a) be a solution of? Justif our answer. Yes, since a a a a must be true (Refleive Prop. of Equalit).. If c 0, d 0, and (c, d) is a solution of, will (d, c) also be a solution? Justif our answer. Yes; replacing with d, with c gives d c c d ; but if (c, d) is a solution, c d d c. So, b the Smmetric Propert of Equalit, d c c d is true. ue for in ation. Determine whether each function represents eponential growth or deca. a. 0.(). growth b.. deca c. 0.(.0). 5 growth. Suppl the reason for each step in the following solution of an eponential equation. 9 7 riginal equation ( ) ( ) Rewrite each side with a base of. ( ) Power of a Power ( ) Propert of Equalit for Eponential Functions Distributive Propert 0 Subtract from each side. Add to each side. Helping You Remember. ne wa to remember that polnomial functions and eponential functions are different is to contrast the polnomial function and the eponential function. Tell at least three was the are different. Sample answer: In, the variable is a base, but in,the variable is an eponent. The graph of is smmetric with respect to the -ais, but the graph of is not. The graph of touches the -ais at (0, 0), but the graph of has the -ais as an asmptote. You can compute the value of mentall for 00, but ou cannot compute the value of mentall for 00. Lesson 0- Eponential Functions 59

17 Assess pen-ended Assessment Modeling Give students a sheet of grid paper and a length of string. Have students model the graph of the equation. Have them check their model b graphing the equation on a graphing calculator. Getting Read for Lesson 0- PREREQUISITE SKILL In Lesson 0-, students will evaluate arithmic epressions. Because arithmic and eponential functions are inverses of each other, their composites are the identit function. Students must be familiar with compositions of functions in order to evaluate these inverse functions. Use Eercises to determine our students familiarit with composition of functions. Answers 75. For h 0, the graph of is translated h units to the right. For h 0, the graph of is translated h units to the left. For k 0, the graph of is translated k units up. For k 0, the graph of is translated k units down. 80. Standardized Test Practice Graphing Calculator Maintain Your Skills Mied Review See margin for graphs. Getting Read for the Net Lesson 68. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 57A 57D. How does an eponential function describe tournament pla? Include the following in our answer: an eplanation of how ou could use the equation to determine the number of rounds of tournament pla for 8 teams, and an eample of an inappropriate number of teams for tournament pla with an eplanation as to wh this number would be inappropriate. 69. If 8, then A A.0. B 6.. C 6.9. D.0. E GRID IN Suppose ou deposit $500 in an account paing.5% interest compounded semiannuall. Find the dollar value of the account rounded to the nearest penn after 0 ears FAMILIES F GRAPHS Graph each pair of functions on the same screen. Then compare the graphs, listing both similarities and differences in shape, asmptotes, domain, range, and -intercepts. 77. See pp. 57A 57D. 7. and 7. and 7. 5 and 5 7. and 75. Describe the effect of changing the values of h and k in the equation h k. See margin. Solve each equation or inequalit. Check our solutions. (Lesson 9-6) p 6, s 6 p s s, a 5 a 6 a 9 a 9 a, or Identif each equation as a tpe of function. Then graph the equation. (Lesson 9-5) square root greatest integer constant Find the inverse of each matri, if it eists. (Lesson -7) does not eist ENERGY A circular cell must deliver 8 watts of energ. If each square centimeter of the cell that is in sunlight produces 0.0 watt of energ, how long must the radius of the cell be? (Lesson 5-8) about.9 cm PREREQUISITE SKILL Find g[h()] and h[g()]. (To review composition of functions, see Lesson 7-7.) See margin. 87. h() 88. h() 89. h() 5 g() 5 g() g() Chapter 0 Eponential and Logarithmic Relations g[h()] 6; h[g()] 88. g[h()] 6 9; h[g()] 89. g[h()] ; h[g()] 50 Chapter 0 Eponential and Logarithmic Relations

18 Logarithms and Logarithmic Functions Lesson Notes Vocabular arithm arithmic function arithmic equation arithmic inequalit Stud Tip Look Back To review inverse functions, see Lesson 7-8. Evaluate arithmic epressions. Solve arithmic equations and inequalities. is a arithmic scale used to measure sound? Man scientific measurements have such an enormous range of possible values that it makes sense to write them as powers of 0 and simpl keep track of their eponents. For eample, the loudness of sound is measured in units called decibels. The graph shows the relative intensities and decibel measures of common sounds. Sound Relative Intensit Decibels pin drop 0 0 whisper ( feet) normal conversation 00 nois kitchen jet engine The decibel measure of the loudness of a sound is the eponent or arithm of its relative intensit multiplied b 0. LGARITHMIC FUNCTINS AND EXPRESSINS To better understand what is meant b a arithm, let s look at the graph of and its inverse. Since eponential functions are one-to-one, the inverse of eists and is also a function. Recall that ou can graph the inverse of a function b interchanging the and values in the ordered pairs of the function (0, ) The inverse of can be defined as. Notice that the graphs of these two functions are reflections of each other over the line. In general, the inverse of b is b. In b, is called the arithm of. It is usuall written as b and is read equals base b of. 80 (, ) (, 0) 0 0 (, ) 0 0 As the value of decreases, the value of approaches 0. Lesson 0- Logarithms and Logarithmic Functions 5 Focus 5-Minute Check Transparenc 0- Use as a quiz or review of Lesson 0-. Mathematical Background notes are available for this lesson on p. 50C. is a arithmic scale used to measure sound? Ask students: n the number line shown, the scale along the bottom is 0 decibels per tick mark. What do ou notice about the scale along the top for relative intensit? The scale is not uniform; the relative intensit at the first tick mark is 0, at the second it is 00, at the third it is 000, and so on. If ou draw a number line with a uniform scale whose tick marks are labeled from 0 to 0, what number is at the midpoint between 0 to 0? 5 0 Where does the point million appear on our number line? ver close to the point for 0 Where does the point 00 appear on our number line? ver, ver close to the point for 0 What problem arises with tring to represent the relative intensities on a standard number line? Sample answer: The lesser intensities are so close together near 0 on the number line that the are difficult to represent accuratel. Resource Manager Workbook and Reproducible Masters Chapter 0 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 58 Practice, p. 58 Reading to Learn Mathematics, p. 58 Enrichment, p. 58 Assessment, p. 6 Transparencies School-to-Career Masters, p. 9 5-Minute Check Transparenc 0- Answer Ke Transparencies Techno Interactive Chalkboard Lesson - Lesson Title 5

19 Teach LGARITHMIC FUNCTINS AND EXPRESSINS Words Smbols Logarithm with Base b Let b and be positive numbers, b. The arithm of with base b is denoted b and is defined as the eponent that makes the equation b true. Suppose b 0 and b. For 0, there is a number such that b if and onl if b. Teaching Tip After discussing the definition of arithm at the bottom of p. 5, write the equation on the chalkboard and ask students to rewrite the equation with in terms of. Repeat this for the equation. ( ) Now write the equation on the chalkboard and ask students to rewrite this equation with in terms of. This will likel have students stmied. Eplain that the rewritten equation is. In-Class Eamples Power Point Write each equation in eponential form. a. 9 9 b Write each equation in arithmic form. a b. 7 7 Evaluate. 5 Stud Tip Look Back To review composition of functions, see Lesson 7-7. Eample Eample Logarithmic to Eponential Form Write each equation in eponential form. a. 8 0 b Eponential to Logarithmic Form Write each equation in arithmic form. a b You can use the definition of arithm to find the value of a arithmic epression. Eample Evaluate Logarithmic Epressions Evaluate 6. 6 Let the arithm equal. 6 Definition of arithm Propert of Equalit for Eponential Functions So, 6 6. The function b, where b 0 and b, is called a arithmic function.. As shown in the graph on the previous page, this function is the inverse of the eponential function b and has the following characteristics.. The function is continuous and one-to-one.. The domain is the set of all positive real numbers.. The -ais is an asmptote of the graph.. The range is the set of all real numbers. 5. The graph contains the point (, 0). That is, the -intercept is. Since the eponential function f() b and the arithmic function g() b are inverses of each other, their composites are the identit function. That is, f[g()] and g[f()]. f[g()] g[ f()] f( b ) g(b ) b b b b 5 Chapter 0 Eponential and Logarithmic Relations 5 Chapter 0 Eponential and Logarithmic Relations

20 Thus, if their bases are the same, eponential and arithmic functions undo each other. You can use this inverse propert of eponents and arithms to simplif epressions. Eample Inverse Propert of Eponents and Logarithms Evaluate each epression. a b. ( ) b b ( ) b b SLVE LGARITHMIC EQUATINS AND INEQUALITIES A arithmic equation is an equation that contains one or more arithms. You can use the definition of a arithm to help ou solve arithmic equations. Eample 5 Solve n 5. n 5 n 5 Solve a Logarithmic Equation riginal equation Definition of arithm In-Class Eample In-Class Eamples 5 6 Power Point Evaluate each epression. a. 9 9 b. 7 7 ( ) SLVE LGARITHMIC EQUATINS AND INEQUALITIES Solve 8 n. 6 Power Point Solve 6. Check our solution. { 6} n ( ) 5 = n 5 Power of a Power n Simplif. A arithmic inequalit is an inequalit that involves arithms. In the case of inequalities, the following propert is helpful. Logarithmic to Eponential Inequalit Smbols If b, 0, and b, then b. If b, 0, and b, then 0 b. Eamples Stud Tip Special Values If b > 0 and b, then the following statements are true. b b because b b. b 0 because b 0. Eample 6 Solve a Logarithmic Inequalit Solve 5. Check our solution. 5 riginal inequalit 0 5 Logarithmic to eponential inequalit 0 5 Simplif. The solution set is { 0 5}. CHECK Tr 5 to see if it satisfies the inequalit. 5 riginal inequalit 5 5? Substitute 5 for. 5 5 because Lesson 0- Logarithms and Logarithmic Functions 5 Lesson 0- Logarithms and Logarithmic Functions 5

21 In-Class Eamples 7 8 Power Point Solve ( ). Check our solution., Solve 7 ( 8) 7 ( 5). Check our solution. Intervention Students have not covered arithmic functions before and are likel to find them confusing. Epect students to need etra time to absorb the material in this lesson before continuing with the rest of the chapter. New Stud Tip Etraneous Solutions The domain of a arithmic function does not include negative values. For this reason, be sure to check for etraneous solutions of arithmic equations. Use the following propert to solve arithmic equations that have arithms with the same base on each side. Smbols Propert of Equalit for Logarithmic Functions If b is a positive number other than, then b b if and onl if. Eample If 7 7, then. Eample 7 Solve Equations with Logarithms on Each Side Solve 5 (p ) 5 p. Check our solution. 5 (p ) 5 p riginal equation p p p p 0 (p )(p ) 0 Factor. Propert of Equalit for Logarithmic Functions Subtract p from each side. p 0 or p 0 Zero Product Propert CHECK p p Solve each equation. Substitute each value into the original equation. 5 ( ) 5 Substitute for p. 5 5 Simplif. 5 [() ] 5 () Substitute for p. Since 5 () is undefined, is an etraneous solution and must be eliminated. Thus, the solution is. Use the following propert to solve arithmic inequalities that have the same base on each side. Eclude values from our solution set that would result in taking the arithm of a number less than or equal to zero in the original inequalit. Smbols Propert of Inequalit for Logarithmic Functions If b, then b b if and onl if, and b b if and onl if. Eample If 9, then 9. Stud Tip Look back To review compound inequalities, see Lesson -6. Eample 8 This propert also holds for and. Solve Inequalities with Logarithms on Each Side Solve 0 ( ) 0 ( 6). Check our solution. 0 ( ) 0 ( 6) riginal inequalit Divide each side b. Propert of Inequalit for Logarithmic Functions Addition and Subtraction Properties of Inequalities We must eclude from this solution all values of such that 0 or 6 0. Thus, the solution set is and 6 and 5. This compound inequalit simplifies to 5. 5 Chapter 0 Eponential and Logarithmic Relations Differentiated Instruction Visual/Spatial Have students create colorful posters showing several equivalent eponential and arithmic equations, such as 8 and 8. Suggest that students use a different color for each of the digits,, and 8 to help them visualize the relative locations of the digits in the pairs of equations. 5 Chapter 0 Eponential and Logarithmic Relations

22 Concept Check Guided Practice GUIDED PRACTICE KEY Eercises Eamples, 5 6, Application. PEN ENDED Give an eample of an eponential equation and its related arithmic equation. Sample answer: 5 and 5. Describe the relationship between and. The are inverses.. FIND THE ERRR Paul and Scott are solving 9. Paul = 9 = 9 = = Scott = 9 = 9 = 9,68 Who is correct? Eplain our reasoning. Scott; see margin for eplanation. Write each equation in arithmic form Write each equation in eponential form Evaluate each epression Solve each equation or inequalit. Check our solutions ( ) ( ) 5, 6. ( 5) ( 7) 6 7. b 9 SUND For Eercises 8 0, use the following information. USA TDAY Snapshots An equation for loudness L, in decibels, is L 0 0 R, where R Jul th can be loud. Be careful. An sound above 85 decibels has the potential is the relative intensit of the sound. to damage hearing. The noisiest Fourth of Jul activities, in decibels: 8. Solve R to find the relative intensit of a fireworks displa with a loudness of 0 decibels. 0 Fireworks 0-90 Car racing Solve R to find Parades 80-0 the relative intensit of a Yard work 95-5 concert with a loudness of 75 decibels Movies 90-0 Concerts How man times more intense is the fireworks displa than the concert? In other words, find Note: Sounds listed b range of peak levels. the ratio of their intensities Source: National Campaign for Hearing Health B Hilar Wasson and Sam Ward, USA TDAY or about 6,8 times Lesson 0- Logarithms and Logarithmic Functions 55 Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 0. include eamples of how to write arithms in eponential form. include an other item(s) that the find helpful in mastering the skills in this lesson. FIND THE ERRR Review converting arithms to eponential form. Also note that, according to Paul,, which cannot be true. Answer. The value of a arithmic equation, 9, is the eponent of the equivalent eponential equation, and the base of the arithmic epression,, is the base of the eponential equation. Thus 9 or 9,68. nline Lesson Plans USA TDAY Education s nline site offers resources and interactive features connected to each da s newspaper. Eperience TDAY, USA TDAY s dail lesson plan, is available on the site and delivered dail to subscribers. This plan provides instruction for integrating USA TDAY graphics and ke editorial features into our mathematics classroom. Log on to Lesson 0- Logarithms and Logarithmic Functions 55

23 About the Eercises rganization b bjective Logarithmic Functions and Epressions: 6, 66 7 Solve Logarithmic Equations and Inequalities: 7 65 dd/even Assignments Eercises 6 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: odd, 5 59 odd, 7 90 Average: 67 odd, 68, 69, 7 90 Advanced: 66 even, 68 8 (optional: 85 90) All: Practice Quiz ( 0) indicates increased difficult Practice and Appl Homework Help For Eercises See Eamples Etra Practice See page 89. Write each equation in arithmic form Write each equation in eponential form Evaluate each epression (n 5) n ( ) WRLD RECRDS For Eercises 5 and 6, use the information given for Eercises 8 0 to find the relative intensit of each sound. Source: The Guinness Book of Records 5. The loudest animal sounds are the low-frequenc pulses made b blue whales when the communicate. These pulses have been measured up to 88 decibels The loudest insect is the African cicada. It produces a calling song that measures 06.7 decibels at a distance of 50 centimeters Solve each equation or inequalit. Check our solutions c 8 c n p 0 0 p 5. ( 8) ( ) Answers riginal equation and 5 5 () Inverse Propert of Eponents and Logarithms Simplif riginal equation and 6 () Inverse Propert of Eponents and Logarithms 56 Chapter 0 Eponential and Logarithmic Relations [ ( 8)] 0 riginal equation 7 [ ( )] ( ) 0 7 ( ) b b n ( ) 6 ( ) ( 0) ( ) 6. 0 (a 6) 0 a a 6. 7 ( 6) Show that each statement is true See margin [ ( 8)] Inverse Propert of Eponents and Logarithms Inverse Propert of Eponents and Logarithms 0 0 Inverse Propert of Eponents and Logarithms 56 Chapter 0 Eponential and Logarithmic Relations

24 See pp. 57A 57D. 66. a. Sketch the graphs of and on the same aes. b. Describe the relationship between the graphs. 67. a. Sketch the graphs of,, ( ), and ( ). b. Describe this famil of graphs in terms of its parent graph. Stud Guide and Intervention, p. 579 (shown) and p. 580 NAME DATE PERID 0- Stud Guide and Intervention Logarithms and Logarithmic Functions Logarithmic Functions and Epressions Definition of Logarithm Let b and be positive numbers, b. The arithm of with base b is denoted with Base b b and is defined as the eponent that makes the equation b true. The inverse of the eponential function b is the arithmic function b. This function is usuall written as b.. The function is continuous and one-to-one.. The domain is the set of all positive real numbers. Properties of. The -ais is an asmptote of the graph. Logarithmic Functions. The range is the set of all real numbers. 5. The graph contains the point (0, ). EARTHQUAKE For Eercises 68 and 69, use the following information. The magnitude of an earthquake is measured on a arithmic scale called the Richter scale. The magnitude M is given b M 0, where represents the amplitude of the seismic wave causing ground motion. 68. How man times as great is the amplitude caused b an earthquake with a Richter scale rating of 7 as an aftershock with a Richter scale rating of? 0 or 000 as times great 69. How man times as great was the motion caused b the 906 San Francisco earthquake that measured 8. on the Richter scale as that caused b the 00 Bhuj, India, earthquake that measured 6.9? 0. or about 5 times as great Eample Write an eponential equation equivalent to 5. 5 Eample Write a arithmic equation equivalent to Eample Evaluate , so 8 6. Eercises Write each equation in arithmic form Write each equation in eponential form Lesson 0- Evaluate each epression. Earthquake The Loma Prieta earthquake measured 7. on the Richter scale and interrupted the 989 World Series in San Francisco. Source: U.S. Geoical Surve 70. NISE RDINANCE A proposed cit ordinance will make it illegal to create sound in a residential area that eceeds 7 decibels during the da and 55 decibels during the night. How man times more intense is the noise level allowed during the da than at night? 0.7 or about 50 times 7. CRITICAL THINKING The value of 5 is between two consecutive integers. Name these integers and eplain how ou determined them. and ; Sample answer: 5 is between and. 7. CRITICAL THINKING Using the definition of a arithmic function where b, eplain wh the base b cannot equal. All powers of are, so the inverse of is not a function. 7. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 57A 57D. Wh is a arithmic scale used to measure sound? Include the following in our answer: the relative intensities of a pin drop, a whisper, normal conversation, kitchen noise, and a jet engine written in scientific notation, a plot of each of these relative intensities on the scale shown below, and , NAME 579 DATE PERID Practice (Average) Skills Practice, p. 58 and Practice, p. 58 (shown) Logarithms and Logarithmic Functions Write each equation in arithmic form Write each equation in eponential form Evaluate each epression (n ) n. Solve each equation or inequalit. Check our solutions n q 0 0 q 0. 6 ( 8). 6. n 8. b Standardized Test Practice an eplanation as to wh the arithmic scale might be preferred over the scale shown above. 7. What is the equation of the function graphed at the right? B A () () B C D Lesson 0- Logarithms and Logarithmic Functions 57 (, 6) (0, ). 8 ( 7) 8 (7 ) 5. 7 (8 0) 7 ( 6) 6. ( ) 7. SUND Sounds that reach levels of 0 decibels or more are painful to humans. What is the relative intensit of 0 decibels? 0 8. INVESTING Maria invests $000 in a savings account that pas 8% interest compounded annuall. The value of the account A at the end of five ears can be determined from the equation A [000( 0.08) 5 ]. Find the value of A to the nearest dollar. $69 Reading to Learn Mathematics, p NAME 58 DATE PERID Reading to Learn Mathematics Logarithms and Logarithmic Functions Pre-Activit Wh is a arithmic scale used to measure sound? Read the introduction to Lesson 0- at the top of page 5 in our tetbook. How man times louder than a whisper is normal conversation? 0 or 0,000 times Reading the Lesson. a. Write an eponential equation that is equivalent to 8. 8 b. Write a arithmic equation that is equivalent to c. Write an eponential equation that is equivalent to 0. 0 d. Write a arithmic equation that is equivalent to e. What is the inverse of the function 5? 5 f. What is the inverse of the function 0? 0 ELL. Match each function with its graph. a. IV b. I c. II I. II. III. Lesson 0- NAME DATE PERID Enrichment, p Enrichment Musical Relationships The frequencies of notes in a musical scale that are one octave apart are related b an eponential equation. For the eight C notes on a piano, the equation is C n C n, where C n represents the frequenc of note C n.. Find the relationship between C and C. C C. Find the relationship between C and C. C 8C The frequencies of consecutive notes are related b a common ratio r. The general equation is f n f r n.. If the frequenc of middle C is 6.6 ccles per second nc of the net 5.. Indicate whether each of the following statements about the eponential function 5 is true or false. a. The -ais is an asmptote of the graph. true b. The domain is the set of all real numbers. false c. The graph contains the point (5, 0). false d. The range is the set of all real numbers. true e. The -intercept is. false Helping You Remember. An important skill needed for working with arithms is changing an equation between arithmic and eponential forms. Using the words base, eponent, and arithm, describe an eas wa to remember and appl the part of the definition of arithm that sas, b if and onl if b. Sample answer: In these equations, b stands for base. In form, b is the subscript, and in eponential form, b is the number that is raised to a power. A arithm is an eponent, so, which is the in the first equation, becomes the eponent in the second equation. Lesson 0- Logarithms and Logarithmic Functions 57

25 Assess pen-ended Assessment Writing Have students write a step-b-step eplanation of the procedure for solving a arithmic equation such as 8 n 7. Getting Read for Lesson 0- PREREQUISITE SKILL In Lesson 0-, students will evaluate epressions using the properties of arithms. Because these properties are related to eponential properties, students must be familiar with eponential properties when multipling or dividing terms with like bases. Use Eercises to determine our students familiarit with multipling and dividing monomials. Assessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons 0- and 0-. Lesson numbers are given to the right of the eercises or instruction lines so students can review concepts not et mastered. Quiz (Lessons 0- and 0-) is available on p. 6 of the Chapter 0 Resource Masters. Maintain Your Skills Mied Review Getting Read for the Net Lesson P ractice Quiz 75. In the figure at the right, if and z w, 7 then D A C. B D 5. Simplif each epression. (Lesson 0-) b 6 b Solve each equation. Check our solutions. (Lesson 9-6) 79., a 5 a 5 a 9 a a 5a 8 Solve each equation b using the method of our choice. Find eact solutions. (Lesson 6-5) p 5p Simplif each epression. (Lesson 9-) 8. ( )( )( 7) BANKING Donna Bowers has $000 she wants to save in the bank. A certificate of deposit (CD) earns 8% annual interest, while a regular savings account earns % annual interest. Ms. Bowers doesn t want to tie up all her mone in a CD, but she has decided she wants to earn $0 in interest for the ear. How much mone should she put in to each tpe of account? (Hint: Use Cramer s Rule.) (Lesson -) $00, CD; $600, savings PREREQUISITE SKILL Simplif. Assume that no variable equals zero. (To review multipling and dividing monomials, see Lesson 5-.) ( ) (a b) 8a 6 b 88. a n7 a an 6 5z 89. n z5 90. b 7 0 a. Determine whether 5(.) represents eponential growth or deca. (Lesson 0-) growth. Write an eponential function whose graph passes through (0, ) and (, ). (). Write an equivalent arithmic equation for (Lesson 0-) z z w Lessons 0- and 0-. Write an equivalent eponential equation for 9 7. (Lesson 0-) 9 7 Evaluate each epression. (Lesson 0-) Solve each equation or inequalit. Check our solutions. (Lessons 0- and 0-) n n 9 9. ( 6) ( ) 5 ( ) 58 Chapter 0 Eponential and Logarithmic Relations 58 Chapter 0 Eponential and Logarithmic Relations

26 We are often confronted with data for which we need to find an equation that best fits the information. We can find eponential and arithmic functions of best fit using a TI-8 Plus graphing calculator. A Follow-Up of Lesson 0- Modeling Real-World Data: Curve Fitting Eample The population per square mile in the United States has changed dramaticall over a period of ears. The table shows the number of people per square mile for several ears. a. Use a graphing calculator to enter the data and draw a scatter plot that shows how the number of people per square mile is related to the ear. Step Enter the ear into L and the people per square mile into L. KEYSTRKES: See pages 87 and 88 to review how to enter lists. Step Step Be sure to clear the Y list. Use the ke to move the cursor from L to L. Draw the scatter plot. KEYSTRKES: See pages 87 and 88 to review how to graph a scatter plot. U.S. Population Densit Year People per People per Year square mile square mile Source: Northeast-Midwest Institute Make sure that Plot is on, the scatter plot is chosen, Xlist is L, and Ylist is L. Use the viewing window [780, 00] with a scale factor of 0 b [0, 5] with a scale factor of 5. We see from the graph that the equation that best fits the data is a curve. Based on the shape of the curve, tr an eponential model. To determine the eponential equation that best fits the data, use the eponential regression feature of the calculator. KEYSTRKES: STAT 0 nd [L], nd [L] ENTER The equation is.85 0 ( ). [780, 00] scl: 0 b [0, 5] scl: 5 (continued on the net page) Graphing Calculator Investigation Modeling Real-World Data: Curve Fitting 59 Graphing Calculator Investigation A Follow-Up of Lesson 0- Getting Started Turning ff Stat Plots Before Step, students should use the kestrokes nd [STAT PLT] and check that both plot and plot are turned off. Diagnostics Displa Students should have the calculator set to Diagnosticn. To set the calculator for diagnostics, use nd [CATALG], move the cursor down to Diagnosticn, and press ENTER twice. Teach When students begin the eercises, the should clear lists L and L. The should also enter appropriate settings for the graphing window. Point out that the table of data is arranged in two double columns. Suggest that students compare their graphs to the one shown. Have students estimate the population densit in 00 and 050. How soon will the population densit be twice what it was in 000? about 00 If ou have time, consider etending this activit into a discussion of how life in the future will be different as the result of the increasing population densit. Ask students to think about the effect on transportation, housing, crime rates, and so on. You ma wish to team-teach with a social studies teacher. Graphing Calculator Investigation Modeling Real-World Data: Curve Fitting 59

27 Graphing Calculator Investigation Assess In Eercise, make sure students can eplain wh their equation of best fit is a good choice. In Eercise, students answers ma var slightl. When ou discuss Eercise 6, ou ma want to ask for an ideas students have about how to use the calculator to judge the relative merits of various models (quadratic, cubic, quartic, and eponential). Answers. The calculator also reports an r value of Recall that this number is a correlation coefficient that indicates how well the equation fits the data. A perfect fit would be r. Therefore, we can conclude that this equation is a prett good fit for the data. To check this equation visuall, overlap the graph of the equation with the scatter plot. KEYSTRKES: VARS 5 GRAPH b. If this trend continues, what will be the population per square mile in 00? To determine the population per square mile in 00, from the graphics screen, find the value of when 00. KEYSTRKES: nd [CALC] 00 ENTER [780, 00] scl: 0 b [0, 5] scl: 5 [780, 00] scl: 0 b [0, 5] scl: 5. [0, 50] scl: 5 b [0, 00] scl: 0 [0, 50] scl: 5 b [0, 00] scl: 0 The calculator returns a value of approimatel If this trend continues, in 00, there will be approimatel 00.6 people per square mile. Eercises In 985, Erika received $0 from her aunt and uncle for her seventh birthda. Her father deposited it into a bank account for her. Both Erika and her father forgot about the mone and made no further deposits or withdrawals. The table shows the account balance for several ears.. Use a graphing calculator to draw a scatter plot for the data. See margin.. Calculate and graph the curve of best fit that shows how the elapsed time is related to the balance. Use EpReg for this eercise. See margin.. Write the equation of best fit ( ) Elapsed Time (ears). Write a sentence that describes the fit of the graph to the data. This equation is a good fit because r. 5. Based on the graph, estimate the balance in ears. Check this using the CALC value. After ears she will have approimatel $ Do ou think there are an other tpes of equations that would be good models for these data? Wh or wh not? A quadratic equation might be a good model for this eample because the shape is close to a portion of a parabola. Balance 0 $ $.0 0 $56. 5 $ $ $.8 0 $ Chapter 0 Eponential and Logarithmic Relations 50 Chapter 0 Eponential and Logarithmic Relations

28 Properties of Logarithms Lesson Notes Simplif and evaluate epressions using the properties of arithms. Solve arithmic equations using the properties of arithms. In Lesson 5-, ou learned that the product of powers is the sum of their eponents. 9 8 or In Lesson 0-, ou learned that arithms are eponents, so ou might epect that a similar propert applies to arithms. Let s consider a specific case. Does (9 8) 9 8? (9 8) ( ) Replace 9 with and 8 with. ( ) Product or 6 of Powers Inverse propert of eponents and arithms 9 8 Replace 9 with and 8 with. or 6 So, (9 8) 9 8. Words are the properties of eponents and arithms related? Inverse propert of eponents and arithms PRPERTIES F LGARITHMS Since arithms are eponents, the properties of arithms can be derived from the properties of eponents. The eample above and other similar eamples suggest the following propert of arithms. Product Propert of Logarithms The arithm of a product is the sum of the arithms of its factors. Smbols For all positive numbers m, n, and b, where b, b mn b m b n. Eample ()(7) 7 Focus 5-Minute Check Transparenc 0- Use as a quiz or review of Lesson 0-. Mathematical Background notes are available for this lesson on p. 50D. are the properties of eponents and arithms related? Ask students: How do ou know that arithms are eponents? Sample answer: The arithm of a number is equal to the power (or eponent) when the number is rewritten in eponential form. Since (9 8) 79, how could 79 have been used in the justification that (9 8) 9 8? After stating that (9 8) 79, then the statements 79 6 and 6 6 could be used to justif that (9 8) 6. To show that this propert is true, let b m and b n. Then, using the definition of arithm, b m and b n. b b mn b mn Product of Powers b b b mn b mn Propert of Equalit for Logarithmic Functions Inverse Propert of Eponents and Logarithms b m b n b mn Replace with b m and with b n. You can use the Product Propert of Logarithms to approimate arithmic epressions. Lesson 0- Properties of Logarithms 5 Resource Manager Workbook and Reproducible Masters Chapter 0 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 587 Practice, p. 588 Reading to Learn Mathematics, p. 589 Enrichment, p. 590 Assessment, pp. 6, 65 Transparencies 5-Minute Check Transparenc 0- Answer Ke Transparencies Techno Interactive Chalkboard Lesson - Lesson Title 5

29 Teach PRPERTIES F LGARITHMS In-Class Eamples Power Point Teaching Tip When discussing the Product Propert of Logarithms, point out that the arithms used in the eample ( ()(7) 7) show the propert applies to all arithms and not just those that can be simplified. Be sure students did not get this impression from the earlier eample where it was shown that (9 8) 9 8. Use to approimate the value of Teaching Tip Some students ma wonder how the approimation for was determined since on most calculators the button calculates onl arithms of base 0. State that 0, which can be 0 evaluated using a calculator. Stress that this procedure will be formall discussed in Lesson 0-. Use and 6.9 to approimate the value of SUND The sound made b a lawnmower has a relative intensit of 0 9 or 90 decibels. Would the sound of ten lawnmowers running at that same intensit be ten times as loud or 900 decibels? Eplain our reasoning. No; the sound of ten lawnmowers is perceived to be onl 0 decibels louder than the sound of one lawnmower, or 00 decibels. TEACHING TIP The value of R in Eample is determined b finding the ratio of the intensit l of the sound in watts per square meter to the intensit l 0 of 0 watts per square meter. The intensit l 0 corresponds to the threshold of hearing. Thus a formula that relates the intensit of a sound in watts per square meter to its loudness in decibels is L 0 l 0. l More About... Sound Technician Sound technicians produce movie sound tracks in motion picture production studios, control the sound of live events such as concerts, or record music in a recording studio. nline Research For information about a career as a sound technician, visit: careers 0 Eample 5 Chapter 0 Eponential and Logarithmic Relations Use the Product Propert Use.5850 to approimate the value of 8. 8 ( ) Replace 8 with 6 or. Product Propert Inverse Propert of Eponents and Logarithms.5850 or Replace with Thus, 8 is approimatel Recall that the quotient of powers is found b subtracting eponents. The propert for the arithm of a quotient is similar. Words The arithm of a quotient is the difference of the arithms of the numerator and the denominator. Smbols For all positive numbers m, n, and b, where b, b m n b m b n. Eample Eample You will show that this propert is true in Eercise 7. Use the Quotient Propert Use Properties of Logarithms SUND The loudness L of a sound in decibels is given b L 0 0 R, where R is the sound s relative intensit. Suppose one person talks with a relative intensit of 0 6 or 60 decibels. Would the sound of ten people each talking at that same intensit be ten times as loud or 600 decibels? Eplain our reasoning. Let L be the loudness of one person talking. L Let L be the loudness of ten people talking. L 0 0 (0 0 6 ) Then the increase in loudness is L L. L L 0 0 (0 0 6 ) Substitute for L and L. 0( ) () or 0 Subtract. Product Propert Distributive Propert Inverse Propert of Eponents and Logarithms Quotient Propert of Logarithms Use and to approimate. 0 Replace with the quotient Quotient Propert or and Thus, is approimatel.68. CHECK Using the definition of arithm and a calculator,.68. The sound of two people talking is perceived b the human ear to be onl about 0 decibels louder than the sound of one person talking, or 70 decibels. Unlocking Misconceptions Power Propert After ou have discussed the Power Propert of Logarithms on p. 5, clarif that the propert works for arithms because the are equivalent to eponents. Stress that students should not read a statement such as 5 5 and conclude that Chapter 0 Eponential and Logarithmic Relations

30 Recall that the power of a power is found b multipling eponents. The propert for the arithm of a power is similar. Eample Power Propert of Logarithms Given 6.95, approimate the value of Replace 6 with 6. 6 Power Propert (.95) or.585 Replace 6 with.95. SLVE LGARITHMIC EQUATINS You can use the properties of arithms to solve equations involving arithms. Eample 5 Power Propert of Logarithms Words The arithm of a power is the product of the arithm and the eponent. Smbols For an real number p and positive numbers m and b, where b, b m p p b m. You will show that this propert is true in Eercise 50. Solve Equations Using Properties of Logarithms Solve each equation. a riginal equation Power Propert Quotient Propert Propert of Equalit for Logarithmic Functions In-Class Eample In-Class Eample Power Point Given 5 6., approimate the value of SLVE LGARITHMIC EQUATINS Power Point 5 Solve each equation. a b. 8 8 ( ) 6 6 Multipl each side b. Take the cube root of each side. The solution is. Stud Tip Checking Solutions It is wise to check all solutions to see if the are valid since the domain of a arithmic function is not the complete set of real numbers. b. ( 6) ( 6) ( 6) ( 6) riginal equation Product Propert Definition of arithm Subtract 6 from each side. ( 8)( ) 0 Factor. 8 0 or 0 Zero Product Propert 8 Solve each equation. CHECK Substitute each value into the original equation. 8 (8 6) () ( 6) 8 () (8) (8 ) Since () and (8) are 6 undefined, is an etraneous solution and must be eliminated. The onl solution is 8. Lesson 0- Properties of Logarithms 5 Differentiated Instruction Interpersonal Right after discussing Eample 5, have pairs of students rework both parts of the eample together without looking at the solution in the tet. Have the partners take turns eplaining the solution steps to each other. Lesson 0- Properties of Logarithms 5

31 Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 0. summarize the properties of arithms the learned in this lesson. include an other item(s) that the find helpful in mastering the skills in this lesson. Concept Check. properties of eponents. Sample answer: 5; 5 Guided Practice GUIDED PRACTICE KEY Eercises Eamples 6,, 7 0 5,. Name the properties that are used to derive the properties of arithms.. PEN ENDED Write an epression that can be simplified b using two or more properties of arithms. Then simplif it.. FIND THE ERRR Umeko and Clemente are simplifing Umeko = = 7 9 Clemente = = 7 7 or Who is correct? Eplain our reasoning. Umeko; see margin for eplanation. Use 0.60 and 7.77 to approimate the value of each epression Solve each equation. Check our solutions. 7. n a 0 (a ) FIND THE ERRR When discussing the error made b Clemente, remind students that arithms are eponents. Adding as 7 (6 ) is similar to saing that or 5, which students should recognize as being untrue because and are unlike terms. About the Eercises rganization b bjective Properties of Logarithms: 0, 7 6 Solve Logarithmic Equations: dd/even Assignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercise 6 involves research on the Internet or other reference materials. Assignment Guide Basic: 7 odd, odd, 5 0, 7 66 Average: odd, 5, 7 66 Advanced: even, 5, 6, 6 (optional: 6 66) indicates increased difficult Practice and Appl Homework Help For Eercises See Eamples 0,, Etra Practice See page 850. Application 5 Chapter 0 Eponential and Logarithmic Relations Answer MEDICINE For Eercises and, use the following information. The ph of a person s blood is given b ph 6. 0 B 0 C, where B is the concentration of bicarbonate, which is a base, in the blood and C is the concentration of carbonic acid in the blood.. ph 6. 0 B C. Use the Quotient Propert of Logarithms to simplif the formula for blood ph.. Most people have a blood ph of 7.. What is the approimate ratio of bicarbonate to carbonic acid for blood with this ph? 0: Use and to approimate the value of each epression Solve each equation. Check our solutions a t ( 5) n z 0 (z ) 8. 6 (a ) 6 9. (b ) (b ) 0. ( ) ( ) p. Clemente incorrectl applied the product and quotient properties of arithms (6 ) or 7 8 Product Propert of Logarithms (8 ) or 7 9 Quotient Propert of Logarithms 5 Chapter 0 Eponential and Logarithmic Relations

32 5. False; ( ), or 5, and 5 since about 0. kilocalorie per gram 0. about 0.89 kilocalories per gram Star Light The Greek astronomer Hipparchus made the first known cata of stars. He listed the brightness of each star on a scale of to 6, the brightest being. With no telescope, he could onl see stars as dim as the 6th magnitude. Source: NASA Answer Solve for n.. ( ). a n a a. b 8 b n b ( ) CRITICAL THINKING Tell whether each statement is true or false. If true, show that it is true. If false, give a countereample. 5. For all positive numbers m, n, and b, where b, b (m n) b m b n. 6. For all positive numbers m, n,, and b, where b, n b m b (n m) b. See pp. 57A 57D. 7. EARTHQUAKES The great Alaskan earthquake in 96 was about 00 times more intense than the Loma Prieta earthquake in San Francisco in 989. Find the difference in the Richter scale magnitudes of the earthquakes. BILGY For Eercises 80, use the following information. The energ E (in kilocalories per gram molecule) needed to transport a substance from the outside to the inside of a living cell is given b E.( 0 C 0 C ), where C is the concentration of the substance outside the cell and C is the concentration inside the cell. 8. Epress the value of E as one arithm. E. C C 9. Suppose the concentration of a substance inside the cell is twice the concentration outside the cell. How much energ is needed to transport the substance on the outside of the cell to the inside? (Use ) 0. Suppose the concentration of a substance inside the cell is four times the concentration outside the cell. How much energ is needed to transport the substance from the outside of the cell to the inside? SUND For Eercises, use the formula for the loudness of sound in Eample on page 5. Use and A certain sound has a relative intensit of R. B how man decibels does the sound increase when the intensit is doubled?. A certain sound has a relative intensit of R. B how man decibels does the sound decrease when the intensit is halved?. A stadium containing 0,000 cheering people can produce a crowd noise of about 90 decibels. If ever one cheers with the same relative intensit, how much noise, in decibels, is a crowd of 0,000 people capable of producing? Eplain our reasoning. About 95 decibels; see margin for eplanation. STAR LIGHT For Eercises 6, use the following information. The brightness, or apparent magnitude, m of a star or planet is given b the formula L m 6.5 0, where L is the amount L 0 of light coming to Earth from the star or planet and L 0 is the amount of light from a sith magnitude star.. Find the difference in the magnitudes of Sirius and the crescent moon Find the difference in the magnitudes of Saturn and Neptune RESEARCH Use the Internet or other reference to find the magnitude of the dimmest stars that we can now see with ground-based telescopes. about. L 0 0 R, where L is the loudness of the sound in decibels and R is the relative intensit of the sound. Since the crowd increased b a factor of, we assume that the intensit also increases b a factor of. Thus, we need to find the loudness of R. L 0 0 R; L 0( 0 0 R) L R; L 0(0.77) 90; L or about 95 Spirals Consider an angle in standard position with its verte at a point called the pole. Its initial side is on a coordinatized ais called the polar ais. A point P on the terminal side of the angle is named b the polar coordinates (r, ), where r is the directed distance of the point from and is the measure of the angle. Graphs in this sstem ma be drawn on polar coordinate paper such as the kind shown below Moon Sirius The crescent moon is about 00 times brighter than the brightest star, Sirius. Saturn Neptune Saturn, as seen from Earth, is 000 times brighter than Neptune. Lesson 0- Properties of Logarithms 55 NAME DATE PERID 0- Enrichment, p Stud Guide and Intervention, p. 585 (shown) and p. 586 NAME DATE PERID 0- Stud Guide and Intervention Properties of Logarithms Properties of Logarithms Properties of eponents can be used to develop the following properties of arithms. Product Propert For all positive numbers m, n, and b, where b, of Logarithms b mn b m b n. Quotient Propert For all positive numbers m, n, and b, where b, of Logarithms b m n b m b n. Power Propert For an real number p and positive numbers m and b, of Logarithms where b, b m p p b m. Eample Use 8.0 and.69 to approimate the value of each epression. a. 6 6 ( ) Eercises b c ( ) (.69) Use 0. and to evaluate each epression , Use and to evaluate each epression NAME 585 DATE PERID Practice (Average) Skills Practice, p. 587 and Practice, p. 588 (shown) Properties of Logarithms Use and to approimate the value of each epression Solve each equation. Check our solutions n u w 8. 9 (u ) u d (m 5) 0 m (b ) 0 b (t 0) 8 (t ) 8 0. (a ) (a ) (r ) 0 r 0 (r ). ( ) ( ). 0 0 w 5. 8 (n ) 8 (n ) 5. 5 ( 9) (9 5) 6 ( ) 7. 6 ( 5) 6 (7 0) 8 8. (5 ) ( ) (c ) 0 (c ) SUND The loudness L of a sound in decibels is given b L 0 0 R, where R is the sound s relative intensit. If the intensit of a certain sound is tripled, b how man decibels does the sound increase? about.8 db. EARTHQUAKES An earthquake rated at.5 on the Richter scale is felt b man people, and an earthquake rated at.5 ma cause local damage. The Richter scale magnitude reading m is given b m 0, where represents the amplitude of the seismic wave causing ground motion. How man times greater is the amplitude of an earthquake that measures.5 on the Richter scale than one that measures.5? 0 times Reading to Learn Mathematics, p NAME 588 DATE PERID Reading to Learn Mathematics Properties of Logarithms Pre-Activit How are the properties of eponents and arithms related? Read the introduction to Lesson 0- at the top of page 5 in our tetbook. Find the value of 5 5. Find the value of 5 5. Find the value of 5 (5 5). Which of the following statements is true? B Reading the Lesson A. 5 (5 5) ( 5 5) ( 5 5) B. 5 (5 5) ELL. Each of the properties of arithms can be stated in words or in smbols. Complete the statements of these properties in words. a. The arithm of a quotient is the difference of the arithms of the numerator and the denominator. b. The arithm of a power is the product of the arithm of the base and the eponent. c. The arithm of a product is the sum of the arithms of its factors.. State whether each of the following equations is true or false. If the statement is true, name the propert of arithms that is illustrated. a. 0 0 true; Quotient Propert b. 8 false c. 8 9 true; Power Propert d false. The algebraic process of solving the equation ( ) leads to or. Does this mean that both and are solutions of the arithmic equation? Eplain our reasoning. Sample answer: No; is a solution because it checks: ( ). However, because () and ( ) are undefined, is an etraneous solution and must be eliminated. The onl solution is. Helping You Remember. A good wa to remember something is to relate it something ou alread know. Use words to eplain how the Product Propert for eponents can help ou remember the product propert for arithms. Sample answer: When ou multipl two numbers or epressions with the same base, ou add the eponents and keep the same base. Logarithms are eponents, so to find the arithm of a product, ou add the arithms of the factors, keeping the same base. Lesson 0- Properties of Logarithms 55 Lesson 0- Lesson 0-

33 Assess pen-ended Assessment Speaking Ask students to eplain the Product Propert, Quotient Propert, and Power Propert of Logarithms in their own words. Encourage them to use specific eamples for clarification. Getting Read for Lesson 0- PREREQUISITE SKILL Students will use common arithms to solve eponential equations and inequalities in Lesson 0-. The solution techniques involve using the skills the learned when solving arithmic equations and inequalities. Use Eercises 6 66 to determine our students familiarit with solving arithmic equations and inequalities. Assessment ptions Quiz (Lesson 0-) is available on p. 6 of the Chapter 0 Resource Masters. Mid-Chapter Test (Lessons 0- through 0-) is available on p. 65 of the Chapter 0 Resource Masters. Standardized Test Practice Maintain Your Skills Mied Review 7. CRITICAL THINKING Use the properties of eponents to prove the Quotient Propert of Logarithms. See margin. 8. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 57A 57D. How are the properties of eponents and arithms related? Include the following in our answer: eamples like the one shown at the beginning of the lesson illustrating the Quotient Propert and Power Propert of Logarithms, and an eplanation of the similarit between one propert of eponents and its related propert of arithms. 9. Simplif A A 5 B 5 C D 50. SHRT RESPNSE Show that b m p p b m for an real number p and positive number m and b, where b. See margin. Evaluate each epression. (Lesson 0-) Solve each equation or inequalit. Check our solutions. (Lesson 0-) 5. 5n a d 9 d d Determine whether each graph represents an odd-degree polnomial function or an even-degree polnomial function. Then state how man real zeros each function has. (Lesson 7-) 57. odd; 58. even; Simplif each epression. (Lesson 9-) 59. 9ab ab b 60. k a 5kl 0kl k z z 5 6. PHYSICS If a stone is dropped from a cliff, the equation t d represents the time t in seconds that it takes for the stone to reach the ground. If d represents the distance in feet that the stone falls, find how long it would take for a stone to fall from a 50-foot cliff. (Lesson 5-6).06 s Answers 7. Let b m and b n. Then b m and b n. b m b n b m n b b b m n b m n Quotient Propert Getting Read for the Net Lesson 56 Chapter 0 Eponential and Logarithmic Relations Propert of Equalit for Logarithmic Equations Inverse Propert of Eponents and Logarithms b m b n b m n Replace with b m and with b n. PREREQUISITE SKILL Solve each equation or inequalit. Check our solutions. (To review solving arithmic equations and inequalities, see Lesson 0-.) 6. ( ) ( ) Let b m, then b m. (b ) p m p b p m p b b p b m p p b m p Product of Powers Propert of Equalit for Logarithmic Equations Inverse Propert of Eponents and Logarithms p b m b m p Replace with b m. 56 Chapter 0 Eponential and Logarithmic Relations

34 Common Logarithms Lesson Notes Vocabular common arithm Change of Base Formula Stud Tip Techno Nongraphing scientific calculators often require entering the number followed b the function, for eample,. LG Solve eponential equations and inequalities using common arithms. Evaluate arithmic epressions using the Change of Base Formula. CMMN LGARITHMS You have seen that the base 0 arithm function, 0, is used in man applications. Base 0 arithms are called common arithms. Common arithms are usuall written without the subscript 0. Most calculators have a Eample LG 0, 0 ke for evaluating common arithms. Sometimes an application of arithms requires that ou use the inverse of arithms, or eponentiation. 0 Eample Solve Logarithmic Equations Using Eponentiation EARTHQUAKES The amount of energ E, in ergs, that an earthquake releases is related to its Richter scale magnitude M b the equation E.8.5M. The Chilean earthquake of 960 measured 8.5 on the Richter scale. How much energ was released? E.8.5M Write the formula. E.8.5(8.5) Replace M with 8.5. E.55 is a arithmic scale used to measure acidit? The ph level of a substance measures its acidit. A low ph indicates an acid solution while a high ph indicates a basic solution. The ph levels of some common substances are shown. The ph level of a substance is given b ph 0 [H], where H is the substance s hdrogen ion concentration in moles per liter. Another wa of writing this formula is ph [H]. Simplif. 0 E 0.55 Write each side using eponents and base 0. E 0.55 E.55 0 Find Common Logarithms Use a calculator to evaluate each epression to four decimal places. a. KEYSTRKES: LG ENTER.7757 about 0.77 b. 0. KEYSTRKES: LG 0. ENTER about Inverse Propert of Eponents and Logarithms Use a calculator. Acidit of Common Substances Substance Batter acid Sauerkraut Tomatoes Black Coffee Milk Distilled Water Eggs Milk of magnesia ph Level The amount of energ released b this earthquake was about.55 0 ergs. Focus 5-Minute Check Transparenc 0- Use as a quiz or review of Lesson 0-. Mathematical Background notes are available for this lesson on p. 50D. is a arithmic scale used to measure acidit? Ask students: What does ph level measure? the acidit of a substance Where have ou heard of ph levels? Sample answer: in soap and shampoo commercials Distilled water has a neutral ph. That is, it is neither acidic nor basic. Use the chart to determine the ph level for a neutral substance. 7.0 Teach CMMN LGARITHMS Teaching Tip Stress that when the base of a arithm is not shown, the base is assumed to be 0. Lesson 0- Common Logarithms 57 Workbook and Reproducible Masters Chapter 0 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 59 Practice, p. 59 Reading to Learn Mathematics, p. 595 Enrichment, p. 596 Resource Manager Transparencies 5-Minute Check Transparenc 0- Real-World Transparenc 0 Answer Ke Transparencies Techno Interactive Chalkboard Lesson - Lesson Title 57

35 In-Class Eamples Power Point Use a calculator to evaluate each epression to four decimal places. a. 6 about b. 0.5 about EARTHQUAKE Refer to Eample. The San Fernando Valle earthquake of 99 measured 6.6 on the Richter scale. How much energ did this earthquake release? about ergs Teaching Tip After discussing In-Class Eample, have students compare the Richter scale magnitudes of the Chilean and San Fernando Valle earthquakes ; the Chilean magnitude was about 9% greater. Then have them compare the energ released b the Chilean earthquake to the energ released b the San Fernando Valle earthquake ; the Chilean earthquake released more than 6 times as much energ. Point out that these results demonstrate the nonlinear nature of the equation that models the amount of energ released. Solve 5 6. about.56 Solve 7 5. { 5.5} Stud Tip Using Logarithms When ou use the Propert for Logarithmic Functions as in the second step of Eample, this is sometimes referred to as taking the arithm of each side. Eample CHANGE F BASE FRMULA The Change of Base Formula allows ou to write equivalent arithmic epressions that have different bases. Change of Base Formula Smbols For all positive numbers, a, b and n, where a and b, a n l ogb n base b of original number. a base b of old base Eample Solve Eponential Equations Using Logarithms Solve. riginal equation CHECK Eample Propert of Equalit for Logarithmic Functions Power Propert of Logarithms l og Divide each side b Use a calculator..88 The solution is approimatel.88. You can check this answer using a calculator or b using estimation. Since 9 and 7, the value of is between and. In addition, the value of should be closer to than, since is closer to 9 than 7. Thus,.88 is a reasonable solution. Solve Eponential Inequalities Using Logarithms Solve riginal inequalit ( ) 8 5 l og b Propert of Inequalit for Logarithmic Functions Power Propert of Logarithms Distributive Propert Subtract 8 from each side. ( 5 8) 8 Distributive Propert CHECK 8 Divide each side b (0.90) Use a calculator. (0.6990) Test. 5 8 riginal inequalit 5 () 8 () Replace with. 5 8 Simplif. 5 6 Negative Eponent Propert The solution set is { 0.756}. 58 Chapter 0 Eponential and Logarithmic Relations Unlocking Misconceptions Change of Base As ou discuss the Change of Base Formula, point out that the base b that students are changing to does not have to be 0. An base could be used; however, b is most commonl 0 because this allows for the arithms to be evaluated with a calculator. 58 Chapter 0 Eponential and Logarithmic Relations

36 Concept Check. Sample answer: 5 ; 0.07 Guided Practice GUIDED PRACTICE KEY Eercises Eamples 6 7, Application To prove this formula, let a n. a n Definition of arithm b a b n Propert of Equalit for Logarithms b a b n Power Propert of Logarithms l ogb n b a Divide each side b b a. a n l ogb n a Replace with a n. b This formula makes it possible to evaluate a arithmic epression of an base b translating the epression into one that involves common arithms. Eample 5 liter indicates increased difficult Practice and Appl Change of Base Formula Epress 5 in terms of common arithms. Then approimate its value to four decimal places. 5 l og 0 5 Change of Base Formula 0.9 Use a calculator. The value of 5 is approimatel.9.. Name the base used b the calculator LG ke. What are these arithms called? 0; common arithms. PEN ENDED Give an eample of an eponential equation requiring the use of arithms to solve. Then solve our equation.. Eplain wh ou must use the Change of Base Formula to find the value of 7 on a calculator. A calculator is not programmed to find base arithms. Use a calculator to evaluate each epression to four decimal places Solve each equation or inequalit. Round to four decimal places.. {p p.888} n 0 {n n 0.907} 9.. a t 5 t p p Epress each arithm in terms of common arithms. Then approimate its value to four decimal places l og 5 ; l og ; l og 9 l ;.699 og 6. DIET Sandra s doctor has told her to avoid foods with a ph that is less than.5. What is the hdrogen ion concentration of foods Sandra is allowed to eat? Use the information at the beginning of the lesson. at least mole per Use a calculator to evaluate each epression to four decimal places Differentiated Instruction Lesson 0- Common Logarithms 59 Naturalist Have interested students research earthquakes and their Richter scale measurements. Have them calculate the amount of energ released b three earthquakes the found to be of interest. Have students compare the energ released to the amount of destruction caused and share their findings with the class. CHANGE F BASE FRMULA In-Class Eample 5 Practice/Appl Stud Notebook Power Point Epress 8 in terms of common arithms. Then approimate its value to four decimal places ; Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 0. include an eample of an eponential inequalit that the solved, and an eample showing how to use the Change of Base Formula. include an other item(s) that the find helpful in mastering the skills in this lesson. About the Eercises rganization b bjective Common Logarithms: 7, 5 55 Change of Base Formula: 5 50 dd/even Assignments Eercises 7 5 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 7 odd, 5, 7, 5, 56, Average: 7 5 odd, Advanced: 8 5 even, 5 7 (optional: 7 77) Lesson 0- Common Logarithms 59

37 Stud Guide and Intervention, p. 59 (shown) and p. 59 NAME DATE PERID 0- Stud Guide and Intervention Common Logarithms Common Logarithms Base 0 arithms are called common arithms. The epression 0 is usuall written without the subscript as. Use the LG ke on our calculator to evaluate common arithms. The relation between eponents and arithms gives the following identit. Inverse Propert of Logarithms and Eponents 0 Eample Evaluate 50 to four decimal places. Use the LG ke on our calculator. To four decimal places, Eample Solve. riginal equation Propert of Equalit for Logarithms ( ) Power Propert of Logarithms Divide each side b. Subtract from each side. Eercises Multipl each side b. Use a calculator to evaluate each epression to four decimal places Solve each equation or inequalit. Round to four decimal places { 0.859} {.5} NAME 59 DATE PERID Practice (Average) Skills Practice, p. 59 and Practice, p. 59 (shown) Common Logarithms Use a calculator to evaluate each epression to four decimal places Use the formula ph [H] to find the ph of each substance given its concentration of hdrogen ions.. milk: [H] mole per liter acid rain: [H] mole per liter black coffee: [H] mole per liter milk of magnesia: [H].6 0 mole per liter 0.5 Solve each equation or inequalit. Round to four decimal places {.69} 9. 5 a z m 00 {m m.0959} b 7. {b b.8699} a z 8 {z z.6555} 8. 5 w n 5 n 0.97 Epress each arithm in terms of common arithms. Then approimate its value to four decimal places ;.50. ; ; ; ; ; HRTICULTURE Siberian irises flourish when the concentration of hdrogen ions [H] in the soil is not less than mole per liter. What is the ph of the soil in which these irises will flourish? 7.8 or less 0. ACIDITY The ph of vinegar is.9 and the ph of milk is 6.6. How man times greater is the hdrogen ion concentration of vinegar than of milk? about BILGY There are initiall 000 bacteria in a culture. The number of bacteria doubles each hour. The number of bacteria N present after t hours is N 000() t. How long will it take the culture to increase to 50,000 bacteria? about 5.6 h. SUND An equation for loudness L in decibels is given b L 0 R, where R is the sound s relative intensit. An air-raid siren can reach 50 decibels and jet engine noise can reach 0 decibels. How man times greater is the relative intensit of the air-raid siren than that of the jet engine noise? 000 Reading to Learn Mathematics, p NAME 59 DATE PERID Reading to Learn Mathematics Common Logarithms ELL Pre-Activit Wh is a arithmic scale used to measure acidit? Read the introduction to Lesson 0- at the top of page 57 in our tetbook. Which substance is more acidic, milk or tomatoes? tomatoes Reading the Lesson. Rhonda used the following kestrokes to enter an epression on her graphing calculator: LG 7 ) ENTER The calculator returned the result.089. Which of the following conclusions are correct? a, c, and d a. The base 0 arithm of 7 is about.0. b. The base 7 arithm of 0 is about.0. c. The common arithm of 7 is about.09. Lesson 0- Homework Help For Eercises See Eamples 7,, Etra Practice See page 850. Solve each equation or inequalit. Round to four decimal places {.0860} a {a a.590} 0. p n z l. 8. og n t l og n 5 n {n n.078} l 9. 6 d d p 5 p {p p.980} og l og 8. n n l og.6 Epress each arithm in terms of common arithms. Then approimate its value to four decimal places (.6) Pollution As little as 0.9 milligram per liter of iron at a ph of 5.5 can cause fish to die. Source: Kentuck Water Watch ACIDITY For Eercises 6, use the information at the beginning of the lesson to find the ph of each substance given its concentration of hdrogen ions.. ammonia: [H] 0 mole per liter. vinegar: [H] 6. 0 mole per liter. 5. lemon juice: [H] mole per liter. 6. orange juice: [H].6 0 mole per liter.5 For Eercises 5 and 5, use the information presented at the beginning of the lesson. 5. PLLUTIN The acidit of water determines the toic effects of runoff into streams from industrial or agricultural areas. A ph range of 6.0 to 9.0 appears to provide protection for freshwater fish. What is this range in terms of the water s hdrogen ion concentration? between and mole per liter 5. BUILDING DESIGN The 97 Slmar earthquake in Los Angeles had a Richter scale magnitude of 6.. Suppose an architect has designed a building strong enough to withstand an earthquake 50 times as intense as the Slmar quake. Find the magnitude of the strongest quake this building is designed to withstand. 8 ASTRNMY For Eercises 5 55, use the following information. Some stars appear bright onl because the are ver close to us. Absolute magnitude M is a measure of how bright a star would appear if it were 0 parsecs, about light ears, awa from Earth. A lower magnitude indicates a brighter star. Absolute magnitude is given b M m 5 5 d, where d is the star s distance from Earth measured in parsecs and m is its apparent magnitude. 5. Sirius 5. Sirius and Vega are two of the brightest stars in Earth s sk. The apparent magnitude of Sirius is. and of Vega is 0.0. Which star appears brighter? 5. Sirius is.6 parsecs from Earth while Vega is 7.76 parsecs from Earth. Find the absolute magnitude of each star. Sirius:.5, Vega: Which star is actuall brighter? That is, which has a lower absolute magnitude? Vega 56. CRITICAL THINKING a. Without using a calculator, find the value of 8 and 8. ; b. Without using a calculator, find the value of 9 7 and 7 9. ; c. Make and prove a conjecture as to the relationship between a b and b a. See margin. 550 Chapter 0 Eponential and Logarithmic Relations d is ver close to 7. e. The common arithm of 7 is eactl Match each epression from the first column with an epression from the second column that has the same value. a. iv i. b. iii ii. 8 c. i iii. 0 d. 5 5 v iv. 5 5 e. 000 ii v. 0.. Calculators do not have kes for finding base 8 arithms directl. However, ou can use a calculator to find 8 0 if ou appl the change of base formula. Which of the following epressions are equal to 8 0? B and C A B. C. D. Helping You Remember. Sometimes it is easier to remember a formula if ou can state it in words. State the change of base formula in words. Sample answer: To change the arithm of a number from one base to another, divide the of the original number in the old base b the of the new base in the old base. 0- Enrichment, p. 596 The Slide Rule Before the invention of electronic calculators, computations were often performed on a slide rule. A slide rule is based on the idea of arithms. It has two movable rods labeled with C and D scales. Each of the scales is arithmic. C D To multipl on a slide rule, move the C rod to the right as shown below. You can find b adding to, and the slide rule adds the lengths for ou. The distance ou get is 0.778, or the arithm of 6. D NAME DATE PERID C Chapter 0 Eponential and Logarithmic Relations

38 Standardized Test Practice MNEY For Eercises 57 and 58, use the following information. If ou deposit P dollars into a bank account paing an annual interest rate r (epressed as a decimal), with n interest paments each ear, the amount A ou r would have after t ears is A P n nt. Marta places $00 in a savings account earning 6% annual interest, compounded quarterl. 57. If Marta adds no more mone to the account, how long will it take the mone in the account to reach $5? about.75 r or r 9 mo 58. How long will it take for Marta s mone to double? about.6 r or r 8 mo 59. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. Wh is a arithmic scale used to measure acidit? Include the following in our answer: the hdrogen ion concentration of three substances listed in the table, and an eplanation as to wh it is important to be able to distinguish between a hdrogen ion concentration of mole per liter and mole per liter. 60. QUANTITATIVE CMPARISIN Compare the quantit in Column A and the quantit in Column B. Then determine whether: A A the quantit in Column A is greater, B the quantit in Column B is greater, C the two quantities are equal, or D the relationship cannot be determined from the information given. Column A Column B If, then what is the value of? C A 0.6 B. C.5 D Assess pen-ended Assessment Writing Ask students to eplain in writing what it means to use the Change of Base Formula. The should include comments about wh this formula is useful. Getting Read for Lesson 0-5 PREREQUISITE SKILL In Lesson 0-5, students will solve eponential equations and inequalities using natural arithms and the skills the learned solving common arithmic equations and inequalities. Students should be confident when converting between eponential and arithmic equations before proceeding. Use Eercises 7 77 to determine our students familiarit with converting between eponential and arithmic equations. Maintain Your Skills Mied Review 66. z 0 z 6 Getting Read for the Net Lesson Use and to approimate the value of each epression. (Lesson 0-) Solve each equation or inequalit. Check our solutions. (Lesson 0-) 65. r z 67. ( 5) Use snthetic substitution to find f() for f() 6. (Lesson 7-) Factor completel. If the polnomial is not factorable, write prime. (Lesson 5-) 69. d d pq 5p 8q 5 7. z z k (d )(d ) (7p )(6q 5) prime PREREQUISITE SKILLS Write an equivalent eponential equation. (For review of arithmic equations, see Lesson 0-.) Write an equivalent arithmic equation. (For review of arithmic equations, see Lesson 0-.) b b Lesson 0- Common Logarithms 55 Answers 56c. conjecture: a b ; b a proof: a b riginal statement b a b b b a b a b a b a Change of Base Formula Inverse Propert of Eponents and Logarithms 59. Comparisons between substances of different acidities are more easil distinguished on a arithmic scale. Answers should include the following. Sample answer: Tomatoes: mole per liter Milk: mole per liter Eggs: mole per liter Those measurements correspond to ph measurements of 5 and, indicating a weak acid and a stronger acid. n the arithmic scale we can see the difference in these acids, whereas on a normal scale, these hdrogen ion concentrations would appear nearl the same. For someone who has to watch the acidit of the foods the eat, this could be the difference between an enjoable meal and heartburn. Lesson 0- Common Logarithms 55

39 Graphing Calculator Investigation A Follow-Up of Lesson 0- Getting Started Using Parentheses In Step of Eample, remind students that the must also use parentheses around the fraction. Teach Before discussing Eample, use a simple equation such as 6 to show students how the equation can be solved b graphing. Graph the equations and 6 and then identif the point of intersection of the graphs. Ask students wh it is necessar in Step to enter the equations using parentheses around the eponents. Have students substitute the solution to Eample into the original equation to verif that it is correct. In Eample, make sure students understand wh the equations must be rewritten using the Change of Base Formula. Students can find the solution set for Eample without using the shading options. Simpl have them use the intersect feature, noting that the graph of Y intersects or is above the graph of Y at and to the right of 0.5. Eample Solve 9 b graphing. Graph each side of the equation. Graph each side of the equation as a separate function. Enter 9 as Y. Enter as Y. Be sure to include the added parentheses around each eponent. Then graph the two equations. KEYSTRKES: See pages 87 and 88 to review graphing equations. The TI-8 Plus has 0 as a built-in function. Enter LG X,T,,n GRAPH to view this graph. To graph arithmic functions with bases other than 0, ou must use the Change of Base Formula, a n l ogb n. b a For eample, l og 0, so to graph 0 ou must enter LG X,T,,n ) LG ) as Y. A Follow-Up of Lesson 0- Solving Eponential and Logarithmic Equations and Inequalities You can use a TI-8 Plus graphing calculator to solve eponential and arithmic equations and inequalities. This can be done b graphing each side of the equation separatel and using the intersect feature on the calculator. [, 8] scl: b [, 8] scl: Use the intersect feature. You can use the intersect feature on the CALC menu to approimate the ordered pair of the point at which the curves cross. KEYSTRKES: See page 5 to review how to use the intersect feature. [, 8] scl: b [, 8] scl: The calculator screen shows that the -coordinate of the point at which the curves cross is. Therefore, the solution of the equation is. 0 [, 8] scl: b [5, 5] scl: 55 Chapter 0 Eponential and Logarithmic Relations 55 Chapter 0 Eponential and Logarithmic Relations

40 Eample Solve b graphing. Rewrite the problem as a sstem of common arithmic inequalities. The first inequalit is or. The second inequalit is. Use the Change of Base Formula to create equations that can be entered into the calculator. l og Thus, the two inequalities are l og and. Enter the first inequalit. Enter l og as Y. Since the inequalit includes less than, shade below the curve. KEYSTRKES: LG LG ) X,T,,n ) Use the arrow and ENTER kes to choose the shade below icon,. Assess In Eercise 9, check that students record the inequalities in the solution set correctl. In particular, students must include the fact that must be greater than 0. Enter the second inequalit. Enter as Y. Since the inequalit includes greater than, shade above the curve. KEYSTRKES: LG X,T,,n ) LG ) GRAPH Use the arrow and ENTER kes to choose the shade above icon,. KEYSTRKES: Graph the inequalities. GRAPH [, 8] scl: b [5, 5] scl: The values of the points in the region where the shadings overlap is the solution set of the original inequalit. Using the calculator s intersect feature, ou can conclude that the solution set is { 0.5}. Eercises Solve each equation or inequalit b graphing ( ) ( 5) ( 7) ( ) 0 Graphing Calculator Investigation Solving Eponential and Logarithmic Equations and Inequalities 55 Graphing Calculator Investigation Solving Eponential and Logarithmic Equations and Inequalities 55

41 Lesson Notes Base e and Natural Logarithms Focus 5-Minute Check Transparenc 0-5 Use as a quiz or review of Lesson 0-. Mathematical Background notes are available for this lesson on p. 50D. is the natural base e used in banking? Ask students: What is the formula calculating? the amount of mone in the account Wh does the interest increase as the time between compounding periods decreases? Sample answer: Interest is earned not just on the initial $ but also on the total interest that has accrued. As the compounding occurs more often, the amount of mone earning interest grows faster. Vocabular natural base, e natural base eponential function natural arithm natural arithmic function Stud Tip Simplifing Epressions with e You can simplif epressions involving e in the same manner in which ou simplif epressions involving. Eamples: 5 e e e 5 Evaluate epressions involving the natural base and natural arithms. Solve eponential equations and inequalities using natural arithms. is the natural base e used in banking? Suppose a bank compounds interest on accounts continuousl, that is, with no waiting time between interest paments. In order to develop an equation to determine continuousl compounded interest, eamine what happens to the value A of an account for increasingl larger numbers of compounding periods n. Use a principal P of $, an interest rate r of 00% or, and time t of ear. BASE e AND NATURAL LGARITHMS In the table above, as n increases, the epression n n() or n n approaches the irrational number This number is referred to as the natural base, e. An eponential function with base e is called a natural base eponential function. The graph of e is shown at the right. Natural base eponential functions are used etensivel in science to model quantities that grow and deca continuousl. Most calculators have an e function for evaluating natural base epressions. Eample Evaluate Natural Base Epressions Continuousl Compounded Interest A=P( + ) nt n A (earl) (quarterl) (monthl) 65 (dail) 8760 (hourl) (+ ) () e e e 0 Use a calculator to evaluate each epression to four decimal places. a. e KEYSTRKES: nd [e ] ENTER about 7.89 b. e. KEYSTRKES: nd [e ]. ENTER.7579 about 0.75 r n (+ ) () (+ ) () 65 (+ ) 65() 8760 (+ ) 8760() (0, )... (, e) e The arithm with base e is called the natural arithm, sometimes denoted b e, but more often abbreviated ln. The natural arithmic function, ln, is the inverse of the natural base eponential function, e. The graph of these two functions shows that ln 0 and ln e. e (, e) (0, ) (e, ) (, 0) ln 55 Chapter 0 Eponential and Logarithmic Relations Resource Manager Workbook and Reproducible Masters Chapter 0 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 599 Practice, p. 600 Reading to Learn Mathematics, p. 60 Enrichment, p. 60 Assessment, p. 6 Science and Mathematics Lab Manual, pp. 7 Transparencies 5-Minute Check Transparenc 0-5 Answer Ke Transparencies Techno AlgePASS: Tutorial Plus, Lesson 9 Interactive Chalkboard

42 Most calculators have an LN ke for evaluating natural arithms. Eample You can write an equivalent base e eponential equation for a natural arithmic equation and vice versa b using the fact that ln e. Eample Since the natural base function and the natural arithmic function are inverses, these two functions can be used to undo each other. e ln ln e EQUATINS AND INEQUALITIES WITH e AND ln Equations and inequalities involving base e are easier to solve using natural arithms than using common arithms. All of the properties of arithms that ou have learned appl to natural arithms as well. Eample 5 Solve 5e 7. 5e 7 5e 9 Solve Base e Equations riginal equation Add 7 to each side. e 9 Divide each side b 5. 5 ln e ln 9 5 ln 9 5 ln Propert of Equalit for Logarithms Inverse Propert of Eponents and Logarithms Divide each side b. Use a calculator. The solution is about CHECK Write Equivalent Epressions Write an equivalent eponential or arithmic equation. a. e 5 b. ln 0.69 e 5 e 5 ln 0.69 e 0.69 ln 5 e 0.69 Eample Evaluate Natural Logarithmic Epressions Use a calculator to evaluate each epression to four decimal places. a. ln KEYSTRKES: LN ENTER.8696 about.86 b. ln 0.05 KEYSTRKES: LN 0.05 ENTER about.9957 Inverse Propert of Base e and Natural Logarithms Evaluate each epression. a. e ln 7 b. ln e e ln 7 7 ln e You can check this value b substituting into the original equation or b finding the intersection of the graphs of 5e 7 and. Teach BASE e AND NATURAL LGARITHMS Teaching Tip Stress that e is a constant like, and not a variable like or. In-Class Eamples Evaluate each epression. a. e ln In-Class Eample 5 Power Point Use a calculator to evaluate each epression to four decimal places. a. e 0.5 about.687 b. e 8 about Use a calculator to evaluate each epression to four decimal places. a. ln about.0986 b. ln about.86 Write an equivalent eponential or arithmic equation. a. e ln b. ln.58 e.58 b. ln e EQUATINS AND INEQUALITIES WITH e AND ln Solve e Power Point Lesson 0-5 Base e and Natural Logarithms 555 Lesson 0-5 Base e and Natural Logarithms 555

43 In-Class Eamples Power Point 6 SAVINGS Suppose ou deposit $700 into an account paing 6% annual interest, compounded continuousl. a. What is the balance after 8 ears? $.5 b. How long will it take for the balance in our account to reach at least $000? at least 7.5 ears 7 Solve each equation or inequalit. a. ln 0.5 about b. ln ( ) Stud Tip Continuousl Compounded Interest Although no banks actuall pa interest compounded continuousl, the equation A = Pe rt is so accurate in computing the amount of mone for quarterl compounding, or dail compounding, that it is often used for this purpose. Stud Tip Equations with ln As with other arithmic equations, remember to check for etraneous solutions. When interest is compounded continuousl, the amount A in an account after t ears is found using the formula A Pe rt, where P is the amount of principal and r is the annual interest rate. Eample 6 Solve Base e Inequalities SAVINGS Suppose ou deposit $000 in an account paing 5% annual interest, compounded continuousl. a. What is the balance after 0 ears? A Pe rt Continuous compounding formula 000e (0.05)(0) Replace P with 000, r with 0.05, and t with e 0.5 Simplif Use a calculator. The balance after 0 ears would be $68.7. b. How long will it take for the balance in our account to reach at least $500? The balance is at least $500. A 500 Write an inequalit. 000e (0.05)t 500 Replace A with 000e (0.05)t. e (0.05)t.5 Divide each side b 000. ln e (0.05)t ln.5 Propert of Equalit for Logarithms 0.05t ln.5 Inverse Propert of Eponents and Logarithms t ln Divide each side b t 8. Use a calculator. It will take at least 8. ears for the balance to reach $500. Eample 7 Solve Natural Log Equations and Inequalities Solve each equation or inequalit. a. ln 5 ln 5 riginal equation e ln 5 e Write each side using eponents and base e. 5 e Inverse Propert of Eponents and Logarithms e 5 Divide each side b Use a calculator. The solution is Check this solution using substitution or graphing. b. ln ( ) ln ( ) riginal inequalit e ln ( ) e Write each side using eponents and base e. e Inverse Propert of Eponents and Logarithms e Add to each side..5 Use a calculator. The solution is all numbers greater than about.5. Check this solution using substitution. 556 Chapter 0 Eponential and Logarithmic Relations Differentiated Instruction Kinesthetic Using plastic coins and paper currenc, have pairs of students begin with $0, choose an interest rate, and calculate how much the will have after 5, 0, 5, and 0 ears. After each calculation, have students model the amount with their mone to help them visualize the growth over time. 556 Chapter 0 Eponential and Logarithmic Relations

44 Concept Check. Elsu; Colb tried to write each side as a power of 0. Since the base of the natural arithmic function is e, he should have written each side as a power of e; 0 ln. Guided Practice GUIDED PRACTICE KEY Eercises Eamples 7, 8, 9 0, , 9 5 Application 8. h 6,00 ln P 0. indicates increased difficult Practice and Appl Homework Help For See Eercises Eamples 0 9, , 5 Etra Practice See page Name the base of natural arithms. the number e. PEN ENDED Give an eample of an eponential equation that requires using natural arithms instead of common arithms to solve. Sample answer:. FIND THE ERRR Colb and Elsu are solving ln 5. e 8 Colb ln = 5 0 ln = 0 5 = 00,000 = 5,000 Who is correct? Eplain our reasoning. Elsu ln = 5 e ln = e 5 = e 5 = e Use a calculator to evaluate each epression to four decimal places.. e e ln ln Write an equivalent eponential or arithmic equation. 8. e ln 9. ln 0 e 0 Evaluate each epression. 0. e ln. ln e 5 5 Solve each equation or inequalit e 0.0. e e ln 6 6. ln ln ALTITUDE For Eercises 8 and 9, use the following information. The altimeter in an airplane gives the altitude or height h (in feet) of a plane above sea level b measuring the outside air pressure P (in kilopascals). The height and air pressure are related b the model P 0. e h 6, Find a formula for the height in terms of the outside air pressure. 9. Use the formula ou found in Eercise 8 to approimate the height of a plane above sea level when the outside air pressure is 57 kilopascals. about 5,066 ft Use a calculator to evaluate each epression to four decimal places. 0. e e e e ln ln ln ln SAVINGS If ou deposit $50 in a savings account paing % interest compounded continuousl, how much mone will ou have after 5 ears? Use the formula presented in Eample 6. $8. 9. PHYSICS The equation ln I0 0.0d relates the intensit of light at a depth of I d centimeters of water I with the intensit in the atmosphere I 0. Find the depth of the water where the intensit of light is half the intensit of the light in the atmosphere. about 9.5 cm Lesson 0-5 Base e and Natural Logarithms 557 Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 0. include eamples of how to evaluate epressions containing natural arithms. include an other item(s) that the find helpful in mastering the skills in this lesson. FIND THE ERRR Make sure students can identif what Colb did incorrectl. Point out that raising both terms to base 0 is not incorrect but the step that follows incorrectl states that 0 ln. About the Eercises rganization b bjective Base e and Natural Logarithms: 0 7 Equations and Inequalities with e and ln: 8 6 dd/even Assignments Eercises 0 5 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 5 odd, 5 59, 6 80 Average: 5 odd, 5 59, 6 80 Advanced: 0 5 even, 5 7 (optional: 75 80) All: Practice Quiz ( 5) Lesson 0-5 Base e and Natural Logarithms 557

45 Stud Guide and Intervention, p. 597 (shown) and p. 598 NAME DATE PERID 0-5 Stud Guide and Intervention Base e and Natural Logarithms Base e and Natural Logarithms The irrational number e.788 often occurs as the base for eponential and arithmic functions that describe real-world phenomena. Natural Base e As n increases, ln e n approaches e.788. The functions e and ln are inverse functions. n Write an equivalent eponential or arithmic equation. 0. e 5. e 6. ln e. ln 5. ln 5 ln 6 e e e 5. Evaluate each epression.. e ln e ln 6. ln e 7. ln e 5 5 Inverse Propert of Base e and Natural Logarithms e ln ln e Natural base epressions can be evaluated using the e and ln kes on our calculator. Eample Evaluate ln 685. Use a calculator. ln Eample Write a arithmic equation equivalent to e 7. e 7 e 7 or ln 7 Eample Evaluate ln e 8. Use the Inverse Propert of Base e and Natural Logarithms. ln e 8 8 Eercises Use a calculator to evaluate each epression to four decimal places.. ln 7. ln 8,50. ln ln ln ln ln 8,5 8. ln Write an equivalent eponential or arithmic equation. 9. e 5 0. e 5. ln 0. ln 8 ln 5 ln 5 e 0 e 8. e ln () e ln ln 0. e 9.6 ln 0 8. e Evaluate each epression. ln 7. ln e 8. e 9. e ln ln e NAME 597 DATE PERID Practice (Average) Skills Practice, p. 599 and Practice, p. 600 (shown) Base e and Natural Logarithms Use a calculator to evaluate each epression to four decimal places.. e ln ln..6. e e ln 0 7. e ln Write an equivalent eponential or arithmic equation. 9. ln ln 6. ln ln e 50 e 6 e e e 8. e e 6. e ln 8 5 ln 0 ln ln ( ) Evaluate each epression. 7. e ln 8. e ln 9. ln e 0. ln e Solve each equation or inequalit.. e 9. e. e.. e 5.8 {.97} e 6. 5e 7 7. e 9 8. e { 0.8}.708 { 0.055} 9. e 8 0. e 5. e e e 55. e e 0 6. e { 0.69} 7. ln 8. ln () 7 9. ln ln ( 6) ln ( ). ln ( ) 5. ln ln 9. ln 5 ln Lesson 0-5 Mone To determine the doubling time on an account paing an interest rate r that is compounded annuall, investors use the Rule of 7. Thus, the amount of time needed for the mone in an account paing 6% interest compounded annuall to double is 7 or 6 ears. Source: Solve each equation or inequalit. 8. e e e e e e e e ln ln ln ( ) ln ( 7) ln ln ln ln ln ( ) ln ln 8, 6 5. ln ln ( ) ln 5 MNEY For Eercises 5 57, use the formula for continuousl compounded interest found in Eample t 00 rin 5. If ou deposit $00 in an account paing.5% interest compounded continuousl, how long will it take for our mone to double? about 9.8 r 55. Suppose ou deposit A dollars in an account paing an interest rate r as a percent, compounded continuousl. Write an equation giving the time t needed for our mone to double, or the doubling time. 56. Eplain wh the equation ou found in Eercise 55 might be referred to as the Rule of In MAKE A CNJECTURE State a rule that could be used to approimate the amount of time t needed to triple the amount of mone in a savings account paing r percent interest compounded continuousl. t 0 r PPULATIN For Eercises 58 and 59, use the following information. In 000, the world s population was about 6 billion. If the world s population continues to grow at a constant rate, the future population P, in billions, can be predicted b P 6e 0.0t, where t is the time in ears since about 7. billion 58. According to this model, what will the world s population be in 00? 59. Some eperts have estimated that the world s food suppl can support a population of, at most, 8 billion. According to this model, for how man more ears will the world s population remain at 8 billion or less? about 55 r nline Research Data Update What is the current world population? Visit to learn more. INVESTING For Eercises 5 and 6, use the formula for continuousl compounded interest, A Pe rt, where P is the principal, r is the annual interest rate, and t is the time in ears. 5. If Sarita deposits $000 in an account paing.% annual interest compounded continuousl, what is the balance in the account after 5 ears? $ How long will it take the balance in Sarita s account to reach $000? about 0. r 7. RADIACTIVE DECAY The amount of a radioactive substance that remains after t ears is given b the equation ae kt, where a is the initial amount present and k is the deca constant for the radioactive substance. If a 00, 50, and k 0.05, find t. about 9.8 r Reading to Learn Mathematics, p NAME 600 DATE PERID Reading to Learn Mathematics Base e and Natural Logarithms Pre-Activit How is the natural base e used in banking? Read the introduction to Lesson 0-5 at the top of page 55 in our tetbook. Suppose that ou deposit $675 in a savings account that pas an annual interest rate of 5%. In each case listed below, indicate which method of compounding would result in more mone in our account at the end of one ear. a. annual compounding or monthl compounding monthl b. quarterl compounding or dail compounding dail c. dail compounding or continuous compounding continuous Reading the Lesson ELL. Jagdish entered the following kestrokes in his calculator: LN 5 ) ENTER The calculator returned the result Which of the following conclusions are correct? d and f 558 Chapter 0 Eponential and Logarithmic Relations RUMRS For Eercises 60 and 6, use the following information. The number of people H who have heard a rumor can be approimated b H, (P PS)e 0.5t where P is the total population, S is the number of people who start the rumor, and t is the time in minutes. Suppose two students start a rumor that the principal will let everone out of school one hour earl that da. 60. If there are 600 students in the school, how man students will have heard the rumor after 0 minutes? about students 6. How much time will pass before half of the students have heard the rumor? about min 6. CRITICAL THINKING Determine whether the following statement is sometimes, alwas, or never true. Eplain our reasoning. Alwas; see pp. 57A-57D. For all positive numbers and, l og l n. ln a. The common arithm of 5 is about.609. b. The natural arithm of 5 is eactl c. The base 5 arithm of e is about.609. d. The natural arithm of 5 is about e is ver close to 5. f. e is ver close to 5.. Match each epression from the first column with its value in the second column. Some choices ma be used more than once or not at all. a. e ln 5 IV I. b. ln V II.0 c. e ln e VI III. d. ln e 5 IV IV. 5 e. ln e I V. 0 f. ln III VI. e e Helping You Remember. A good wa to remember something is to eplain it to someone else. Suppose that ou are studing with a classmate who is puzzled when asked to evaluate ln e. How would ou eplain to him an eas wa to figure this out? Sample answer: ln means natural. The natural of e is the power to which ou raise e to get e. This is obviousl. 0-5 Enrichment, p. 60 Approimations for and e The following epression can be used to approimate e. If greater and greater values of n are used, the value of the epression approimates e more and more closel. n n NAME DATE PERID Another wa to approimate e is to use this infinite sum. The greater the value of n, the closer the approimation. e n In a similar manner, can be approimated using an infinite product discovered b the English mathematician John Wallis (66 70) n n n n Solve each problem.. Use a cal ator with an e ke to find e to 7 decimal places Chapter 0 Eponential and Logarithmic Relations

46 Standardized Test Practice Maintain Your Skills P Mied Review 66. l og l og Getting Read for the Net Lesson ractice Quiz 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How is the natural base e used in banking? Include the following in our answer: an eplanation of how to calculate the value of an account whose interest is compounded continuousl, and an eplanation of how to use natural arithms to find when the account will have a specified value. 6. If e and e what is the value of? B, A. B 0.5 C.00 D SHRT RESPNSE The population of a certain countr can be modeled b the equation P(t) 0 e 0.0t, where P is the population in millions and t is the number of ears since 900. When will the population be 00 million, 00 million, and 00 million? What do ou notice about these time periods? 96, 98, 05; It takes between and 5 ears for the population to double. Epress each arithm in terms of common arithms. Then approimate its value to four decimal places. (Lesson 0-) Solve each equation. Check our solutions. (Lesson 0-) 69. (a ) (a ) State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. (Lesson 9-) 7. mn inverse, 7. b a c joint, 7. 7 direct, 7 7. CMMUNICATIN A microphone is placed at the focus of a parabolic reflector to collect sounds for the television broadcast of a football game. The focus of the parabola that is the cross section of the reflector is 5 inches from the verte. The latus rectum is 0 inches long. Assuming that the focus is at the origin and the parabola opens to the right, write the equation of the cross section. (Lesson 8-) PREREQUISITE SKILL Solve each equation or inequalit. 0 5 (To review eponential equations and inequalities, see Lesson 0-.) ( 0.) ( 0.5) ( 0.) Epress 5 in terms of common arithms. Then approimate its value to four decimal places. (Lesson 0-) l og 5 ;.60. Write an equivalent eponential equation for ln. (Lesson 0-5) e Lessons 0- through 0-5 Assess pen-ended Assessment Speaking Ask students to eplain how evaluating epressions involving base e and natural arithms is similar to evaluating epressions involving common arithms and base 0, and also how the differ. Getting Read for Lesson 0-6 PREREQUISITE SKILL Students will encounter eponential growth and deca problems in Lesson 0-6. The will be required to solve eponential equations and inequalities. Use Eercises to determine our students familiarit with solving eponential equations and inequalities. Assessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons 0- through 0-5. Lesson numbers are given to the right of the eercises or instruction lines so students can review concepts not et mastered. Quiz (Lessons 0- and 0-5) is available on p. 6 of the Chapter 0 Resource Masters. Solve each equation or inequalit. (Lesson 0- through 0-5). (9 5) ( ) e 7.86 Lesson 0-5 Base e and Natural Logarithms 559 Answer 6. The number e is used in the formula for continuousl compounded interest, A Pe rt. Although no banks actuall pa interest compounded continuall, the equation is so accurate in computing the amount of mone for quarterl compounding, or dail compounding, that it is often used for this purpose. Answers should include the following. If ou know the annual interest rate r and the principal P, the value of the account after t ears is calculated b multipling P times e raised to the r times t power. Use a calculator to find the value of e rt. If ou know the value A ou wish the account to achieve, the principal P, and the annual interest rate r, the time t needed to achieve this value is found b first taking the natural arithm of A minus the natural arithm of P. Then, divide this quantit b r. Lesson 0-5 Base e and Natural Logarithms 559

47 Lesson Notes Eponential Growth and Deca Focus 5-Minute Check Transparenc 0-6 Use as a quiz or review of Lesson 0-5. Mathematical Background notes are available for this lesson on p. 50D. can ou determine the current value of our car? Ask students: What kinds of items increase in value? Sample answer: artwork, some trading cards, some collectibles During which ear does the car depreciate the most? first ear How can the amount of depreciation be different each ear when the percent of decrease is alwas the same? In the first ear, the car depreciates 6% of its value at the beginning of that ear (when it was new). This is when its value is greatest, so the amount of depreciation is also the greatest during this ear. The amount of depreciation decreases each ear because the value of the car at the beginning of each ear is less than it was at the beginning of the previous ear. Vocabular rate of deca rate of growth Stud Tip Rate of Change Remember to rewrite the rate of change as a decimal before using it in the formula. Use arithms to solve problems involving eponential deca. Use arithms to solve problems involving eponential growth. Certain assets, like homes, can appreciate or increase in value over time. thers, like cars, depreciate or decrease in value with time. Suppose ou bu a car for $,000 and the value of the car decreases b 6% each ear. The table shows the value of the car each ear for up to 5 ears after it was purchased. EXPNENTIAL DECAY The depreciation of the value of a car is an eample of eponential deca. When a quantit decreases b a fied percent each ear, or other period of time, the amount of that quantit after t ears is given b a( r) t, where a is the initial amount and r is the percent of decrease epressed as a decimal. The percent of decrease r is also referred to as the rate of deca. Eample Eponential Deca of the Form a( r) t CAFFEINE A cup of coffee contains 0 milligrams of caffeine. If caffeine is eliminated from the bod at a rate of % per hour, how long will it take for half of this caffeine to be eliminated from a person s bod? Eplore Plan can ou determine the current value of our car? The problem gives the amount of caffeine consumed and the rate at which the caffeine is eliminated. It asks ou to find the time it will take for half of the caffeine to be eliminated from a person s bod. Use the formula a( r) t. Let t be the number of hours since drinking the coffee. The amount remaining is half of 0 or 65. Solve a( r) t Eponential deca formula 65 0( 0.) t Replace with 65, a with 0, and r with % or (0.89) t Divide each side b (0.89) t 0.5 t (0.89) Propert of Equalit for Logarithms Product Propert for Logarithms 0. 5 t Divide each side b l og t Use a calculator. Years after Purchase 0 5 Value of Car ($), , ,5.0,09.9 0, Chapter 0 Eponential and Logarithmic Relations Resource Manager Workbook and Reproducible Masters Chapter 0 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 605 Practice, p. 606 Reading to Learn Mathematics, p. 607 Enrichment, p. 608 Assessment, p. 6 Graphing Calculator and Spreadsheet Masters, p. 6 School-to-Career Masters, p. 0 Teaching Algebra With Manipulatives Masters, p. 78 Transparencies 5-Minute Check Transparenc 0-6 Answer Ke Transparencies Techno Interactive Chalkboard

48 Paleontoist Paleontoists stud fossils found in geoical formations. The use these fossils to trace the evolution of plant and animal life and the geoic histor of Earth. nline Research For information about a career as a paleontoist, visit: careers Source: U.S. Department of Labor Eamine It will take approimatel 6 hours for half of the caffeine to be eliminated from a person s bod. Use the formula to find how much of the original 0 milligrams of caffeine would remain after 6 hours. a( r) t Eponential deca formula 0( 0.) 6 Replace a with 0, r with 0., and t with Use a calculator. Half of 0 is 65, so the answer seems reasonable. Another model for eponential deca is given b ae kt, where k is a constant. This is the model preferred b scientists. Use this model to solve problems involving radioactive deca. Eample Eponential Deca of the Form ae kt PALENTLGY The half-life of a radioactive substance is the time it takes for half of the atoms of the substance to become disintegrated. All life on Earth contains the radioactive element Carbon-, which decas continuousl at a fied rate. The half-life of Carbon- is 5760 ears. That is, ever 5760 ears half of a mass of Carbon- decas awa. a. What is the value of k for Carbon-? To determine the constant k for Carbon-, let a be the initial amount of the substance. The amount that remains after 5760 ears is then represented b a or 0.5a. ae kt Eponential deca formula 0.5a ae k(5760) Replace with 0.5a and t with e 5760k Divide each side b a. ln 0.5 ln e 5760k Propert of Equalit for Logarithmic Functions ln k Inverse Propert of Eponents and Logarithms ln 0.5 k 5760 Divide each side b k Use a calculator. The constant for Carbon- is Thus, the equation for the deca of Carbon- is ae 0.000t, where t is given in ears. b. A paleontoist eamining the bones of a wooll mammoth estimates that the contain onl % as much Carbon- as the would have contained when the animal was alive. How long ago did the mammoth die? Let a be the initial amount of Carbon- in the animal s bod. Then the amount that remains after t ears is % of a or 0.0a. ae 0.000t Formula for the deca of Carbon- 0.0a ae 0.000t Replace with 0.0a. 0.0 e 0.000t Divide each side b a. ln 0.0 ln e 0.000t Propert of Equalit for Logarithms ln t Inverse Propert of Eponents and Logarithms ln 0.0 t Divide each side b , t Use a calculator. The mammoth lived about 9,000 ears ago. Teach EXPNENTIAL DECAY Teaching Tip In Eample, point out that ou are calculating how long until half the caffeine has been eliminated, which also means half the caffeine remains. If the value to be found is something other than half, students must be careful that the use the formula correctl. In-Class Eamples Power Point CAFFEINE Refer to Eample. How long will it take for 90% of this caffeine to be eliminated from a person s bod? about 0 h GELGY The half-life of Sodium- is.6 ears. a. What is the value of k for Sodium-? about b. A geoist eamining a meteorite estimates that it contains onl about 0% as much Sodium- as it would have contained when it reached Earth s surface. How long ago did the meteorite reach the surface of Earth? about 9 ears ago Lesson 0-6 Eponential Growth and Deca 56 Differentiated Instruction Logical Have students work in pairs or small groups. Ask them to eamine the growth and deca formulas used in Eamples and to discuss how the equations are related. In particular, ask them to discuss how the can identif which equations are used for eponential deca situations (minus/negative sign) and which are used for eponential growth. Lesson 0-6 Eponential Growth and Deca 56

49 EXPNENTIAL GRWTH In-Class Eamples Power Point The population of a cit of one million is increasing at a rate of % per ear. If the population continues to grow at this rate, in how man ears will the population have doubled? D A ears B 5 ears C 0 ears D ears PPULATIN As of 000, Nigeria had an estimated population of 7 million people and the United States had an estimated population of 78 million people. The populations of Nigeria and the United States can be modeled b N(t) 7e 0.06t and U(t) 78e 0.009t, respectivel. According to these models, when will Nigeria s population be more than the population of the United States? after 6 ears or in 06 Standardized Test Practice Test-Taking Tip To change a percent to a decimal, drop the percent smbol and move the decimal point two places to the left..5% 0.05 EXPNENTIAL GRWTH When a quantit increases b a fied percent each time period, the amount of that quantit after t time periods is given b a( r) t, where a is the initial amount and r is the percent of increase epressed as a decimal. The percent of increase r is also referred to as the rate of growth. Eample Multiple-Choice Test Item Eponential Growth of the Form a( r) t In 90, the population of a cit was 0,000. Since then, the population has increased b eactl.5% per ear. If the population continues to grow at this rate, what will the population be in 00? A 8,000 B 5,85 C,06,690 D. 0 Read the Test Item You need to find the population of the cit or 00 ears later. Since the population is growing at a fied percent each ear, use the formula a( r) t. Solve the Test Item a( r) t Eponential growth formula 0,000( 0.05) 00 Replace a = 0,000, r with 0.05, and t with or 00. 0,000(.05) 00 5,85.8 The answer is B. Simplif. Use a calculator. Another model for eponential growth, preferred b scientists, is ae kt, where k is a constant. Use this model to find the constant k. Eample Eponential Growth of the Form ae kt PPULATIN As of 000, China was the world s most populous countr, with an estimated population of.6 billion people. The second most populous countr was India, with.0 billion. The populations of India and China can be modeled b I(t).0e 0.05t and C(t).6e 0.009t, respectivel. According to these models, when will India s population be more than China s? You want to find t such that I(t) C(t). I(t) C(t).0e 0.05t.6e 0.009t Replace I(t) with.0e 0.05t and C(t) with.6e 0.009t ln.0e 0.05t ln.6e 0.009t ln.0 ln e 0.05t ln.6 ln e 0.009t Propert of Inequalit for Logarithms Product Propert of Logarithms ln t ln t Inverse Propert of Eponents and Logarithms 0.006t ln.6 ln.0 Subtract from each side. t ln.6 ln.0 Divide each side b t 6.86 Use a calculator. After 7 ears or in 07, India will be the most populous countr in the world. 56 Chapter 0 Eponential and Logarithmic Relations 56 Chapter 0 Eponential and Logarithmic Relations Standardized Test Practice Eample n all standardized tests, students should look to identif an answer choices that can be icall eliminated. In Eample, students can quickl determine that since % of 0,000 is 00 and therefore.5% is 800, over the 00 ears from 90 to 00 the cit s population will have increased b more than 800(00) or 80,000 people. So Choice A is much too low. Choice D can also be eliminated, because. 0 written in standard notation is 0,000,000,000 (which is 0 billion). That s more than the population of the entire planet! So, the answer must be either Choice B or Choice C.

50 Concept Check Guided Practice GUIDED PRACTICE KEY Eercises Eamples 6 7, 8 9, indicates increased difficult Practice and Appl Homework Help For Eercises Standardized Test Practice See Eamples 0,, 7 0 5, 6 Etra Practice See page 85.. more than,000 ears ago PPULATIN GRWTH For Eercises 7 and 8, use the following information. The cit of Raleigh, North Carolina, grew from a population of,000 in 990 to a population of 59,000 in Write an eponential growth equation of the form ae kt for Raleigh, where t is the number of ears after 990.,000e 0.05t 8. Use our equation to predict the population of Raleigh in 00. about 9,59 people 9. Suppose the weight of a bar of soap decreases b.5% each time it is used. If the bar weighs 95 grams when it is new, what is its weight to the nearest gram after 5 uses? C A 57.5 g B 59. g C 65 g D 9 g Write a general formula for eponential a( r) t growth and deca where r is the percent of change., where r 0 represents eponential growth and r 0 represents. Eplain how to solve ( r) t for t. See margin. eponential deca. PEN ENDED Give an eample of a quantit that grows or decas at a fied rate. Sample answer: mone in a bank SPACE For Eercises 6, use the following information. A radioisotope is used as a power source for a satellite. The power output P (in watts) is given b P 50e t 50, where t is the time in das.. Is the formula for power output an eample of eponential growth or deca? Eplain our reasoning. Deca; the eponent is negative. 5. Find the power available after 00 das. about.5 watts 6. Ten watts of power are required to operate the equipment in the satellite. How long can the satellite continue to operate? about 0 das 0. CMPUTERS Zeus Industries bought a computer for $500. It is epected to depreciate at a rate of 0% per ear. What will the value of the computer be in ears? $600. REAL ESTATE The Martins bought a condominium for $85,000. Assuming that the value of the condo will appreciate at most 5% a ear, how much will the condo be worth in 5 ears? at most $08,8.9. MEDICINE Radioactive iodine is used to determine the health of the throid gland. It decas according to the equation ae t, where t is in das. Find the half-life of this substance. about 8. das. PALENTLGY A paleontoist finds a bone that might be a dinosaur bone. In the laborator, she finds that the Carbon- found in the bone is of that found in living bone tissue. Could this bone have belonged to a dinosaur? Eplain our reasoning. (Hint: The dinosaurs lived from 0 million ears ago to 6 million ears ago.) No; the bone is onl about,000 ears old, and dinosaurs died out 6,000,000 ears ago.. ANTHRPLGY An anthropoist finds there is so little remaining Carbon- in a prehistoric bone that instruments cannot measure it. This means that there is less than 0.5% of the amount of Carbon- the bones would have contained when the person was alive. How long ago did the person die? Lesson 0-6 Eponential Growth and Deca 56 Practice/Appl Stud Notebook Have students complete the definitions/eamples for the remaining terms on their Vocabular Builder worksheets for Chapter 0. record the formulas for eponential growth and deca. include an other item(s) that the find helpful in mastering the skills in this lesson. About the Eercises rganization b bjective Eponential Deca: 0 0 Eponential Growth: 0 0 Assignment Guide Basic:,, 7, 8, 0 Average:,, 5 8, 0 Advanced: 0 even, 5, 6, 9 0 Answer. Take the common arithm of each side, use the Power Propert to write ( r) t as t ( r), and then divide each side b the quantit ( r). Lesson 0-6 Eponential Growth and Deca 56

51 Stud Guide and Intervention, p. 60 (shown) and p. 60 NAME DATE PERID 0-6 Stud Guide and Intervention Eponential Growth and Deca Eponential Deca Depreciation of value and radioactive deca are eamples of eponential deca. When a quantit decreases b a fied percent each time period, the amount of the quantit after t time periods is given b a( r) t, where a is the initial amount and r is the percent decrease epressed as a decimal. Another eponential deca model often used b scientists is ae kt, where k is a constant. Eample CNSUMER PRICES As techno advances, the price of man technoical devices such as scientific calculators and camcorders goes down. ne brand of hand-held organizer sells for $89. a. If its price decreases b 6% per ear, how much will it cost after 5 ears? Use the eponential deca model with initial amount $89, percent decrease 0.06, and time 5 ears. a( r) t Eponential deca formula 89( 0.06) 5 a 89, r 0.06, t 5 $65. After 5 ears the price will be $65.. b. After how man ears will its price be $50? To find when the price will be $50, again use the eponential deca formula and solve for t. a( r) t Eponential deca formula 50 89( 0.06) t 50, a 89, r (0.9) t Divide each side b (0.9) t t 0.9 Power Propert Propert of Equalit for Logarithms t Divide each side b t 9. The price will be $50 after about 9. ears. Eercises Lesson 0-6 lmpics The women s high jump competition first took place in the USA in 895, but it did not become an lmpic event until 98. Source: BILGY For Eercises 5 and 6, use the following information. Bacteria usuall reproduce b a process known as binar fission. In this tpe of reproduction, one bacterium divides, forming two bacteria. Under ideal conditions, some bacteria reproduce ever 0 minutes. 5. about Find the constant k for this tpe of bacteria under ideal conditions. 6. Write the equation for modeling the eponential growth of this bacterium. ae 0.07t ECNMICS For Eercises 7 and 8, use the following information. The annual Gross Domestic Product (GDP) of a countr is the value of all of the goods and services produced in the countr during a ear. During the period , the Gross Domestic Product of the United States grew about.% per ear, measured in 996 dollars. In 985, the GDP was $577 billion. 7. Assuming this rate of growth continues, what will the GDP of the United States be in the ear 00? $,565 billion 8. In what ear will the GDP reach $0 trillion? about 05. BUSINESS A furniture store is closing out its business. Each week the owner lowers prices b 5%. After how man weeks will the sale price of a $500 item drop below $00? 6 weeks CARBN DATING Use the formula ae 0.000t, where a is the initial amount of Carbon-, t is the number of ears ago the animal lived, and is the remaining amount after t ears.. How old is a fossil remain that has lost 95% of its Carbon-? about 5,000 ears old. How old is a skeleton that has 95% of its Carbon- remaining? about 7.5 ears old 0-6 NAME 60 DATE PERID Practice (Average) Skills Practice, p. 605 and Practice, p. 606 (shown) Eponential Growth and Deca Solve each problem. r t. INVESTING The formula A P gives the value of an investment after t ears in an account that earns an annual interest rate r compounded twice a ear. Suppose $500 is invested at 6% annual interest compounded twice a ear. In how man ears will the investment be worth $000? about.7 r 9. LYMPICS In 98, when the high jump was first introduced as a women s sport at the lmpic Games, the winning women s jump was 6.5 inches, while the winning men s jump was 76.5 inches. Since then, the winning jump for women has increased b about 0.8% per ear, while the winning jump for men has increased at a slower rate, 0.%. If these rates continue, when will the women s winning high jump be higher than the men s? after the ear 8 0. HME WNERSHIP The Mendes famil bought a new house 0 ears ago for $0,000. The house is now worth $9,000. Assuming a stead rate of growth, what was the earl rate of appreciation?.7%. BACTERIA How man hours will it take a culture of bacteria to increase from 0 to 000 if the growth rate per hour is 85%? about 7.5 h. RADIACTIVE DECAY A radioactive substance has a half-life of ears. Find the constant k in the deca formula for the substance. about DEPRECIATIN A piece of machiner valued at $50,000 depreciates at a fied rate of % per ear. After how man ears will the value have depreciated to $00,000? about 7. r 5. INFLATIN For Dave to bu a new car comparabl equipped to the one he bought 8 ears ago would cost $,500. Since Dave bought the car, the inflation rate for cars like his has been at an average annual rate of 5.%. If Dave originall paid $800 for the car, how long ago did he bu it? about 8 r 6. RADIACTIVE DECAY Cobalt, an element used to make allos, has several isotopes. ne of these, cobalt-60, is radioactive and has a half-life of 5.7 ears. Cobalt-60 is used to trace the path of nonradioactive substances in a sstem. What is the value of k for Cobalt-60? about WHALES Modern whales appeared 50 million ears ago. The vertebrae of a whale discovered b paleontoists contain roughl 0.5% as much carbon- as the would have contained when the whale was alive. How long ago did the whale die? Use k about 50,000 r 8. PPULATIN The population of rabbits in an area is modeled b the growth equation P(t) 8e 0.6t, where P is in thousands and t is in ears. How long will it take for the population to reach 5,000? about. r 9. DEPRECIATIN A computer sstem depreciates at an average rate of % per month. If the value of the computer sstem was originall $,000, in how man months is it worth $750? about mo 0. BILGY In a laborator, a culture increases from 0 to 95 organisms in 5 hours. What is the hourl growth rate in the growth formula a( r) t? about 5.% Reading to Learn Mathematics, p NAME 606 DATE PERID Reading to Learn Mathematics Eponential Growth and Deca ELL Pre-Activit How can ou determine the current value of our car? Read the introduction to Lesson 0-6 at the top of page 560 in our tetbook. Between which two ears shown in the table did the car depreciate b the greatest amount? between ears 0 and Describe two was to calculate the value of the car 6 ears after it was purchased. (Do not actuall calculate the value.) Sample answer:. Multipl $ b 0.6 and subtract the result from $ Multipl $ b 0.8. Reading the Lesson. State whether each situation is an eample of eponential growth or deca. a. A cit had,000 residents in 980 and 8,000 residents in 000. growth b. Raul compared the value of his car when he bought it new to the value when he traded ;lpit in si ears later. deca Standardized Test Practice 56 Chapter 0 Eponential and Logarithmic Relations. CRITICAL THINKING The half-life of Radium is 60 ears. When will a 0-gram sample of Radium be completel gone? Eplain our reasoning. Never; theoreticall, the amount left will alwas be half of the previous amount.. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can ou determine the current value of our car? Include the following in our answer: a description of how to find the percent decrease in the value of the car each ear, and a description of how to find the value of a car for an given ear when the rate of depreciation is known.. SHRT RESPNSE An artist creates a sculpture out of salt that weighs 000 pounds. If the sculpture loses.5% of its mass each ear to erosion, after how man ears will the statue weigh less than 000 pounds? about 9.5 r. The curve shown at the right represents a portion of the graph of which function? D A 50 B C e D 5 c. A paleontoist compared the amount of carbon- in the skeleton of an animal when it died to the amount 00 ears later. deca d. Maria deposited $750 in a savings account paing.5% annual interest compounded quarterl. She did not make an withdrawals or further deposits. She compared the balance in her passbook immediatel after she opened the account to the balance ears later. growth. State whether each equation represents eponential growth or deca. a. 5e 0.5t growth b. 000( 0.05) t deca c. 0.e 00t deca d. ( 0.000) t growth Helping You Remember. Visualizing their graphs is often a good wa to remember the difference between mathematical equations. How can our knowledge of the graphs of eponential equations from Lesson 0- help ou to remember that equations of the form a( r) t represent eponential growth, while equations of the form a( r) t represent eponential deca? Sample answer: If a 0, the graph of ab is alwas increasing if b and is alwas decreasing if 0 b. Since r is alwas a positive number, if b r, the base will be greater than and the function will be increasing (growth), while if b r, the base will be less than and the function will be decreasing (deca). NAME DATE PERID 0-6 Enrichment, p. 608 Effective Annual Yield When interest is compounded more than once per ear, the effective annual ield is higher than the annual interest rate. The effective annual ield, E, is the interest rate that would give the same amount of interest if the interest were compounded once per ear. If P dollars are invested for one ear, the value of the investment at the end of the ear is A P( E). If P dollars are invested for one ear at a nominal rate r compounded n times per ear, r the value of the investment at the end of the ear is A P n n. Setting the amounts equal and solving for E will produce a formula for the effective annual ield. P( E) P n r n E n r n E n r n 56 Chapter 0 Eponential and Logarithmic Relations If compound s continuous, the value of the investment at the end of one set the

52 Maintain Your Skills Mied Review Write an equivalent eponential or arithmic equation. (Lesson 0-5) 5. e ln 6. e n 9 7. ln ln 8 ln 9 n e 8 Solve each equation or inequalit. Round to four decimal places. (Lesson 0-) p 000 p.9 0. b 8 9 BUSINESS For Eercises, use the following information. The board of a small corporation decided that 8% of the annual profits would be divided among the si managers of the corporation. There are two sales managers and four nonsales managers. Fift percent of the amount would be split equall among all si managers. The other 50% would be split among the four nonsales managers. Let p represent the annual profits of the corporation. (Lesson 9-). Write an epression to represent the share of the profits each nonsales manager will receive. p 0.5(0.08p) 0.5(0.08p). Simplif this epression Write an epression in simplest form to represent the share of the profits each sales manager will receive. p 50 Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hperbola. (Lesson 8-6) hperbola ellipse 6. 8 parabola circle AGRICULTURE For Eercises 8 0, use the graph at the right. (Lesson 5-) 8. Write the number of pounds of pecans produced b U.S. growers in 000 in scientific notation Write the number of pounds of pecans produced b the state of Georgia in 000 in scientific notation What percent of the overall pecan production for 000 can be attributed to Georgia? about 8.8% USA TDAY Snapshots Georgia led pecan production in 000 U.S. growers produced more than 06 million pounds of pecans in 000. States producing the most pecans (in pounds): Georgia New Meico Teas Louisiana Alabama Arizona million 0 million 7 million 5 million million 80 million Assess pen-ended Assessment Modeling Using manipulatives, ask students to demonstrate wh the amount of compound interest earned annuall increases each ear. Have students relate this to their understanding of eponential growth. Assessment ptions Quiz (Lesson 0-6) is available on p. 6 of the Chapter 0 Resource Masters. Answer. Answers should include the following. Find the absolute value of the difference between the price of the car for two consecutive ears. Then divide this difference b the price of the car for the earlier ear. Find minus the rate of decrease in the value of the car as a decimal. Raise this value to the number of ears it has been since the car was purchased, and then multipl b the original value of the car. Source: National Agricultural Statistics Service B Sam Ward, USA TDAY n Quake Anniversar, Japan Still Worries It is time to complete our project. Use the information and data ou have gathered about earthquakes to prepare a research report or Web page. Be sure to include graphs, tables, diagrams, and an calculations ou need for the earthquake ou chose. Lesson 0-6 Eponential Growth and Deca 565 nline Lesson Plans USA TDAY Education s nline site offers resources and interactive features connected to each da s newspaper. Eperience TDAY, USA TDAY s dail lesson plan, is available on the site and delivered dail to subscribers. This plan provides instruction for integrating USA TDAY graphics and ke editorial features into our mathematics classroom. Log on to Lesson 0-6 Eponential Growth and Deca 565

53 Stud Guide and Review Vocabular and Concept Check This alphabetical list of vocabular terms in Chapter 0 includes a page reference where each term was introduced. Assessment A vocabular test/review for Chapter 0 is available on p. 6 of the Chapter 0 Resource Masters. Lesson-b-Lesson Review For each lesson, the main ideas are summarized, additional eamples review concepts, and practice eercises are provided. Vocabular PuzzleMaker ELL The Vocabular PuzzleMaker software improves students mathematics vocabular using four puzzle formats crossword, scramble, word search using a word list, and word search using clues. Students can work on a computer screen or from a printed handout. MindJogger Videoquizzes ELL MindJogger Videoquizzes provide an alternative review of concepts presented in this chapter. Students work in teams in a game show format to gain points for correct answers. The questions are presented in three rounds. Round Concepts (5 questions) Round Skills ( questions) Round Problem Solving ( questions) Vocabular and Concept Check Change of Base Formula (p. 58) common arithm (p. 57) eponential deca (p. 5) eponential equation (p. 56) eponential function (p. 5) eponential growth (p. 5) eponential inequalit (p. 57) arithm (p. 5) arithmic equation (p. 5) arithmic function (p. 5) arithmic inequalit (p. 5) State whether each sentence is true or false. If false, replace the underlined word(s) to make a true statement.. If, then b the Propert of Equalit for Eponential Functions. true. The number of bacteria in a petri dish over time is an eample of eponential deca. false; eponential growth. The natural arithm is the inverse of the eponential function with base 0.. The Power Propert of Logarithms shows that ln 9 ln If a savings account ields % interest per ear, then % is the rate of growth. 6. Radioactive half-life is an eample of eponential deca. true 7. The inverse of an eponential function is a composite function. 8. The Quotient Propert of Logarithms is shown b. 9. The function f() (5) is an eample of a quadratic function. false; eponential function 0- See pages Eponential Functions Concept Summar An eponential function is in the form ab, where a 0, b 0, and b. The function ab represents eponential growth for a 0 and b, and eponential deca for a 0 and 0 b. Propert of Equalit for Eponential Functions: If b is a positive number other than, then b b if and onl if. Propert of Inequalit for Eponential Functions: If b, then b b if and onl if, and b b if and onl if. 566 Chapter 0 Eponential and Logarithmic Relations For more information about Foldables, see Teaching Mathematics with Foldables. TM natural base, e (p. 55) natural base eponential function (p. 55) natural arithm (p. 55) natural arithmic function (p. 55) Power Propert of Logarithms (p. 5) Product Propert of Logarithms (p. 5) Propert of Equalit for Eponential Functions (p. 56) Propert of Equalit for Logarithmic Functions (p. 5) Propert of Inequalit for Eponential Functions (p. 57) Propert of Inequalit for Logarithmic Functions (p. 5) Quotient Propert of Logarithms (p. 5) rate of deca (p. 560) rate of growth (p. 56). false; common arithm. false; Propert of Inequalit for Logarithms 5. true 7. false; arithmic function 8. false; Product Propert of Logarithms Have students look through the chapter to make sure the have included notes and eamples for each lesson in this chapter in their Foldable. Encourage students to refer to their Foldables while completing the Stud Guide and Review and to use them in preparing for the Chapter Test. 566 Chapter 0 Eponential and Logarithmic Relations

54 Chapter 0 Stud Guide and Review Stud Guide and Review Eample Solve 6 n for n. 6 n riginal equation 6 n Rewrite 6 as 6 so each side has the same base. 6 n Propert of Equalit for Eponential Functions 5 n The solution is 5. Eercises Determine whether each function represents eponential growth or deca. See Eample on page (0.7) deca. () growth Write an eponential function whose graph passes through the given points. See Eample on page 55.. (0, ) and (, 5) (). (0, 7) and (,.) 7 5 Solve each equation or inequalit. See Eamples 5 and 6 on pages 56 and p 7 p or 6 0- See pages Eamples Logarithms and Logarithmic Functions Concept Summar Eamples Suppose b 0 and b. For 0, there is a 7 7 number such that b if and onl if b. Logarithmic to eponential inequalit: 5 5 If b, 0, and b, then b. 0 If b, 0, and b, then 0 b. Propert of Equalit for Logarithmic Functions: If 5 5 6, If b is a positive number other than, then 6. then b b if and onl if. Propert of Inequalit for Logarithmic Functions: If 0, If b, then b b if and onl if, then 0. and b b if and onl if. Solve 9 n. 9 n n 9 riginal inequalit Logarithmic to eponential inequalit n ( ) 9 n Power of a Power n 7 Simplif. Chapter 0 Stud Guide and Review 567 Chapter 0 Stud Guide and Review 567

55 Stud Guide and Review Chapter 0 Stud Guide and Review Solve. riginal equation Propert of Equalit for Logarithmic Functions 6 Divide each side b. Eercises Write each equation in arithmic form. See Eample on page Write each equation in eponential form. See Eample on page Evaluate each epression. See Eamples and on pages 5 and Solve each equation or inequalit. See Eamples 5 8 on pages 5 and b 9. 8 ( ) 8 ( 5). 5 5 (5 ). 8 ( ) 8, 0- See pages Eample Properties of Logarithms Concept Summar The arithm of a product is the sum of the arithms of its factors. The arithm of a quotient is the difference of the arithms of the numerator and the denominator. The arithm of a power is the product of the arithm and the eponent. Use and 8.6 to approimate the value of. 8 Replace with Quotient Propert or 0.79 Replace 9 with 0.88 and 8 with.6. Eercises Use and to approimate the value of each epression. See Eamples and on page Solve each equation. See Eample 5 on page ( ) m Chapter 0 Eponential and Logarithmic Relations 568 Chapter 0 Eponential and Logarithmic Relations

56 Chapter 0 Stud Guide and Review Stud Guide and Review 0- See pages Eample Common Logarithms Concept Summar Base 0 arithms are called common arithms and are usuall written without the subscript 0: 0. You use the inverse of arithms, or eponentiation, to solve equations or inequalities involving common arithms: 0. The Change of Base Formula: a n l og n b base b original number a base b old base Solve riginal equation b Propert of Equalit for Logarithmic Functions Power Propert of Logarithms l og 7 Divide each side b or Use a calculator. Eercises Solve each equation or inequalit. Round to four decimal places. See Eamples and on page Epress each arithm in terms of common arithms. Then approimate its value to four decimal places. See Eample 5 on page l og ; l og 5 ; lo g 000 ; Base e and Natural Logarithms See pages Concept Summar You can write an equivalent base e eponential equation for a natural arithmic equation and vice versa b using the fact that ln e. Since the natural base function and the natural arithmic function are inverses, these two functions can be used to undo each other. e ln and ln e Eample Solve ln ( ) 5. ln ( ) 5 riginal inequalit e ln ( ) e 5 Write each side using eponents and base e. e 5 e 5. Inverse Propert of Eponents and Logarithms Subtract from each side. Use a calculator. Chapter 0 Stud Guide and Review 569 Chapter 0 Stud Guide and Review 569

57 Stud Guide and Review Etra Practice, see pages Mied Problem Solving, see page 87. Eercises Write an equivalent eponential or arithmic equation. See Eample on page e 6 ln 6 5. ln 7. e 7. Evaluate each epression. See Eample on page e ln 55. ln e 7 7 Solve each equation or inequalit. See Eamples 5 and 7 on pages 555 and e e e ln ln ( 0) ln ln See pages Eample Eponential Growth and Deca Concept Summar Eponential deca: a( r) t or ae kt Eponential growth: a( r) t or ae kt BILGY A certain culture of bacteria will grow from 500 to 000 bacteria in.5 hours. Find the constant k for the growth formula. Use = ne kt. ae kt Eponential growth formula e k(.5) Replace with 000, a with 500, and t with.5. 8 e.5k Divide each side b 500. ln 8 ln e.5k ln 8.5k Propert of Equalit for Logarithmic Functions Inverse Propert of Eponents and Logarithms l n 8 k Divide each side b k Use a calculator. Eercises See Eamples on pages BUSINESS Able Industries bought a fa machine for $50. It is epected to depreciate at a rate of 5% per ear. What will be the value of the fa machine in ears? $ BILGY For a certain strain of bacteria, k is 0.87 when t is measured in das. How long will it take 9 bacteria to increase to 78 bacteria? 5.05 das 6. CHEMISTRY Radium-6 decomposes radioactivel. Its half-life, the time it takes for half of the sample to decompose, is 800 ears. Find the constant k in the deca formula for this compound. about PPULATIN The population of a cit 0 ears ago was 5,600. Since then, the population has increased at a stead rate each ear. If the population is currentl 6,800, find the annual rate of growth for this cit. about.6% 570 Chapter 0 Eponential and Logarithmic Relations 570 Chapter 0 Eponential and Logarithmic Relations

58 Practice Test Vocabular and Concepts Choose the term that best completes each sentence.. The equation 0.() is an eponential ( growth, deca) function.. The arithm of a quotient is the (sum, difference ) of the arithms of the numerator and the denominator.. The base of a natural arithm is (0, e). Skills and Applications. Write 7 87 in arithmic form Write 8 6 in eponential form Write an eponential function whose graph passes through (0, 0.) and (, 6.). 0.() 7. Epress 5 in terms of common arithms. l og 5 8. Evaluate. 5 Use 7.07 and to approimate the value of each epression Simplif each epression Solve each equation or inequalit. Round to four decimal places if necessar , p p m ( ) 9 ( ) (8 7) 5 ( 5) 0. ( ) e e ln ln CINS For Eercises 8 and 9, use the following information. You bu a commemorative coin for $5. The value of the coin increases.5% per ear. 8. How much will the coin be worth in 5 ears? $ After how man ears will the coin have doubled in value? 0. QUANTITATIVE CMPARISIN Compare the quantit in Column A and the quantit in Column B. Then determine whether: B A the quantit in Column A is greater, B the quantit in Column B is greater, C the two quantities are equal, or D the relationship cannot be determined from the information given. Column A the current value of the account if the annual interest rate is % compunded quarterl Column B $00 was deposited in an account 5 ears ago. Portfolio Suggestion the current value of the account if the annual interest rate is % compounded continuousl Chapter 0 Practice Test 57 Introduction In mathematics, eponential functions can be used to model real-world problems. The solution to the eponential function provides a solution to the real-world problem. Ask Students Find a real-world problem modeled b an eponential function from our work in this chapter and show how ou solved it. Eplain how the function models the real-world situation and what could be gained b understanding the real-world problem better. Place our work in our portfolio. Assessment ptions Vocabular Test A vocabular test/review for Chapter 0 can be found on p. 6 of the Chapter 0 Resource Masters. Chapter Tests There are si Chapter 0 Tests and an pen- Ended Assessment task available in the Chapter 0 Resource Masters. Chapter 0 Tests Form Tpe Level Pages MC basic A MC average 6 6 B MC average 6 6 C FR average D FR average FR advanced MC = multiple-choice questions FR = free-response questions pen-ended Assessment Performance tasks for Chapter 0 can be found on p. 6 of the Chapter 0 Resource Masters. A sample scoring rubric for these tasks appears on p. A5. Unit Test A unit test/review can be found on pp of the Chapter 0 Resource Masters. TestCheck and Worksheet Builder This networkable software has three modules for assessment. Worksheet Builder to make worksheets and tests. Student Module to take tests on-screen. Management Sstem to keep student records. Chapter 0 Practice Test 57

59 Standardized Test Practice These two pages contain practice questions in the various formats that can be found on the most frequentl given standardized tests. A practice answer sheet for these two pages can be found on p. A of the Chapter 0 Resource Masters. Standardized 0 Standardized Test Practice Practice Student Recording Sheet, p. A NAME DATE PERID Student Record Sheet (Use with pages of the Student Edition.) Part Record our answers on the answer sheet provided b our teacher or on a sheet of paper.. The arc shown is part of a circle. Find 8 the area of the shaded region. B A B C D Multiple Choice 8 units 6 units units 6 units 8 5. ( ) z 0 D z A B C z D 6. If v 6 0, then v 6 v A A 6. B. C. D The epression 5 is equivalent to A Part Multiple Choice Select the best answer from the choices given and fill in the corresponding oval. A B C D A B C D 7 A B C D 9 A B C D A B C D 5 A B C D 8 A B C D 0 A B C D A B C D 6 A B C D Part Short Response/Grid In Solve the problem and write our answer in the blank. For Questions 8, also enter our answer b writing each number or smbol in a bo. Then fill in the corresponding oval for that number or smbol. 5 7 / / / / / / Answers. If line is parallel to line m in the figure below, what is the value of? D 0 50 m A C 0 B D 70 A C 5. B D What are all the values for such that 8? B A C B 6 D 6 9. If f() 8, what are all the values of at which f() 0? B / / / / Part Quantitative Comparison / / / / Select the best answer from the choices given and fill in the corresponding oval. 9 A B C D A B C D A B C D 0 A B C D A B C D Additional Practice See pp in the Chapter 0 Resource Masters for additional standardized test practice.. According to the graph, what was the percent of increase in sales from 998 to 000? D A 5% B 5% C 5% D 50%. What is the -intercept of the line described b the equation 5? B A C 5 Sales ($ thousands) Year 5 0 D 5 B A C 0, B, 0, 6, 0, 6 D,, 0. Which of the following is equal to 7. 5( 0 )? 500( 0 D ) A 0.05(0 ) B 0.5(0 ) C 0.005(0 ) D 0.05(0 ) Test-Taking Tip Question 7 You can use estimates to help ou eliminate answer choices. For eample, in Question 7, ou can estimate that 5 is less than 9, which is 7 or. Eliminate choices C and D. 57 Chapter 0 Standardized Test Practice Log n for Test Practice The Princeton Review offers additional test-taking tips and practice problems at their web site. Visit or TestCheck and Worksheet Builder Special banks of standardized test questions similar to those on the SAT, ACT, TIMSS 8, NAEP 8, and Algebra End-of-Course tests can be found on this CD-RM. 57 Chapter 0 Eponential and Logarithmic Relations

60 Aligned and verified b Part Short Response/Grid In Part Quantitative Comparison Record our answers on the answer sheet provided b our teacher or on a sheet of paper.. If the outer diameter of a clindrical tank is 6.6 centimeters and the inner diameter is 5. centimeters, what is the thickness of the tank?.57 cm 6.6 cm 5. cm Compare the quantit in Column A and the quantit in Column B. Then determine whether: A the quantit in Column A is greater, B the quantit in Column B is greater, C the two quantities are equal, or D the relationship cannot be determined from the information given. Column A 9. 0 Column B Top View D. What number added to 80% of itself is equal to 5? 5. f 00 families surveed, 95% have at least one TV and 60% of those with TVs have more than TVs. If 50 families have eactl TVs, how man families have eactl TV? 6 0. C r z t z s. In the figure, if ED 8, what is the measure of line segment AE? 5 B 0 5. A t B C r s A E D 5. If a b is defined as a b ab, find the value of If 6(m k) 6 (m k), what is the value of m k? 7. If and, what number is found b subtracting the least possible value of from the greatest possible value of? 9 8. If f() ( )( )( e), what is the difference between the greatest and least roots of f()? Round to the nearest hundredth.. C circumference of circle 8 perimeter of square ABCD. z 5 z 9 A 6 z 6. n 0 B A D n ( n) Chapter 0 Standardized Test Practice 57 Chapter 0 Standardized Test Practice 57