Chapter 10 Resource Masters

Transcription

1 Chapter 0 Resource Masters

2 DATE PERIOD 0 Reading to Learn Mathematics Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter 0. As you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term Change of Base Formula Found on Page Definition/Description/Eample Vocabulary Builder common logarithm LAW guh RIH thuhm eponential decay! "" # "" $! "# " $ EHK spuh NEHN chuhl eponential equation eponential function eponential growth eponential inequality (continued on the net page) Glencoe/McGraw-Hill vii Glencoe Algebra 2

3 DATE PERIOD 0 Reading to Learn Mathematics Vocabulary Builder (continued) logarithm Vocabulary Term Found on Page Definition/Description/Eample logarithmic function! "" # "" $ LAW guh RIHTH mihk natural base, e natural base eponential function natural logarithm natural logarithmic function rate of decay rate of growth Glencoe/McGraw-Hill viii Glencoe Algebra 2

4 0- DATE PERIOD Study Guide and Intervention Eponential Functions Eponential Functions An eponential function has the form y ab, where a 0, b 0, and b. Properties of an Eponential Function Eponential Growth and Decay Eample. The function is continuous and one-to-one. 2. The domain is the set of all real numbers.. The -ais is the asymptote 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 y ab represents eponential growth. If a 0 and 0 b, the function y ab represents eponential decay. Sketch the graph of y 0.(). Then state the function s domain and range. Make a table of values. Connect the points to form a smooth curve. y Lesson y The domain of the function is all real numbers, while the range is the set of all positive real numbers. O Eample 2 Determine whether each function represents eponential growth or decay. a. y 0.5(2) b. y 2.8(2) c. y.(0.5) eponential growth, neither, since 2.8, eponential decay, since since the base, 2, 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.. y (2) 2. y 2. y 0.25(5) y y y O O O 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 decay.. y 0.(.2) growth 5. y 5 neither 6. y (0) 5 decay Glencoe/McGraw-Hill 57 Glencoe Algebra 2

5 0- DATE PERIOD Study Guide and Intervention (continued) Eponential Functions Eponential Equations and Inequalities All the properties of rational eponents that you know also apply to real eponents. Remember that a m a n a m n,(a m ) n a mn, and a m a n a m n. Property of Equality for If b is a positive number other than, Eponential Functions then b b y if and only if y. Property of Inequality for Eponential Functions If b then b b y if and only if y and b b y if and only if y. Eample Eample 2 Solve Original equation (2 2 ) 2 5 Rewrite as ( ) 5 Prop. of Inequality for Eponential Functions Distributive Property 7 Subtract and add 2 to each side. Eercises Simplify each epression. Solve Original inequality Rewrite as Add to each side. Divide each side by 2. The solution set is { }. Prop. of Inequality for Eponential Functions. ( 2 ) ( 2 y 2 ) or y 6. ( 6 )( 5 ) 5. ( 6 ) 5 6. (2 )(5 ) Solve each equation or inequality. Check your solution or Glencoe/McGraw-Hill 57 Glencoe Algebra 2

6 0- DATE PERIOD Skills Practice Eponential Functions Sketch the graph of each function. Then state the function s domain and range.. y (2) 2. y 2 2 y y O domain: all real numbers; range: all positive numbers O domain: all real numbers; range: all positive numbers Lesson 0- Determine whether each function represents eponential growth or decay.. y (6) growth. y 2 decay 5. y 0 decay 6. y 2(2.5) growth 9 0 Write an eponential function whose graph passes through the given points. 7. (0, ) and (, ) y 8. (0, ) and (, 2) y () 9. (0, ) and (, 6) y 2 0. (0, 5) and (, 5) y 5(). (0, 0.) and (, 0.5) y 0.(5) 2. (0, 0.2) and (,.6) y 0.2(8) Simplify each epression.. ( ) 27. ( 2 ) Solve each equation or inequality. Check your solution Glencoe/McGraw-Hill 575 Glencoe Algebra 2

7 0- DATE PERIOD Practice (Average) Eponential Functions Sketch the graph of each function. Then state the function s domain and range.. y.5(2) 2. y (). y (0.5) y y O O 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 decay.. y 5(0.6) decay 5. y 0.(2) growth 6. y 5 decay Write an eponential function whose graph passes through the given points. 7. (0, ) and (, ) 8. (0, 2) and (, 0) 9. (0, ) and (,.5) y y 2(5) y (0.5) 0. (0, 0.8) and (,.6). (0, 0.) and (2, 0) 2. (0, ) and (, 8 ) y 0.8(2) y 0.(5) y (2) Simplify each epression.. (2 2 ) 8 6. (n ) 75 n 5 5. y 6 y 5 6 y n n n Solve each equation or inequality. Check your solution n 5 9 n n n n n 2. 6 n 282n BIOLOGY For Eercises 25 and 26, use the following information. The initial number of bacteria in a culture is 2,000. The number after days is 96, Write an eponential function to model the population y of bacteria after days. y 2,000(2) 26. How many bacteria are there after 6 days? 768, EDUCATION A college with a graduating class of 000 students in the year 2002 predicts that it will have a graduating class of 862 in years. Write an eponential function to model the number of students y in the graduating class t years after y 000(.05) t Glencoe/McGraw-Hill 576 Glencoe Algebra 2

8 0- DATE PERIOD Reading to Learn Mathematics Eponential Functions Pre-Activity How does an eponential function describe tournament play? Read the introduction to Lesson 0- at the top of page 52 in your tetbook. How many rounds of play would be needed for a tournament with 00 players? 7 Reading the Lesson. Indicate whether each of the following statements about the eponential function y 0 is true or false. a. The domain is the set of all positive real numbers. false b. The y-intercept is. true c. The function is one-to-one. true d. The y-ais is an asymptote of the graph. false e. The range is the set of all real numbers. false Lesson 0-2. Determine whether each function represents eponential growth or decay. a. y 0.2(). growth b. y. decay c. y 0.(.0). growth. Supply the reason for each step in the following solution of an eponential equation Original equation 2 5 ( 2 ) 2 ( ) Rewrite each side with a base of. 2(2 ) Power of a Power 2(2 ) Property of Equality for Eponential Functions 2 Distributive Property 2 0 Subtract from each side. 2 Add 2 to each side. Helping You Remember. One way to remember that polynomial functions and eponential functions are different is to contrast the polynomial function y 2 and the eponential function y 2.Tell at least three ways they are different. Sample answer: In y 2, the variable is a base, but in y 2, the variable is an eponent. The graph of y 2 is symmetric with respect to the y-ais, but the graph of y 2 is not. The graph of y 2 touches the -ais at (0, 0), but the graph of y 2 has the -ais as an asymptote. You can compute the value of y 2 mentally for 00, but you cannot compute the value of y 2 mentally for 00. Glencoe/McGraw-Hill 577 Glencoe Algebra 2

9 0- DATE PERIOD Enrichment Finding Solutions of y y Perhaps you have noticed that if and y are interchanged in equations such as y and y, the resulting equation is equivalent to the original equation. The same is true of the equation y y. However, finding solutions of y y and drawing its graph is not a simple process. Solve each problem. Assume that and y are positive real numbers.. If a 0, will (a, a) be a solution of y y? Justify your answer. 2. If c 0, d 0, and (c, d) is a solution of y y,will (d, c) also be a solution? Justify your answer.. Use 2 as a value for y in y y. The equation becomes 2 2. a. Find equations for two functions, f() and g() that you could graph to find the solutions of 2 2. Then graph the functions on a separate sheet of graph paper. b. Use the graph you drew for part a to state two solutions for 2 2. Then use these solutions to state two solutions for y y.. In this eercise, a graphing calculator will be very helpful. Use the technique of Eercise to complete the tables below. Then graph y y for positive values of and y. If there are asymptotes, show them in your diagram using dotted lines. Note that in the table, some values of y call for one value of, others call for two. y y O y Glencoe/McGraw-Hill 578 Glencoe Algebra 2

10 0-2 DATE PERIOD Study Guide and Intervention Logarithms and Logarithmic Functions Logarithmic Functions and Epressions Definition of Logarithm with Base b Let b and be positive numbers, b. The logarithm of with base b is denoted log b and is defined as the eponent y that makes the equation b y true. The inverse of the eponential function y b is the logarithmic function b y. This function is usually written as y log b. Properties of Logarithmic Functions. The function is continuous and one-to-one. 2. The domain is the set of all positive real numbers.. The y-ais is an asymptote of the graph.. The range is the set of all real numbers. 5. The graph contains the point (0, ). Eample 5 2 Write an eponential equation equivalent to log 2 5. Eample 2 log 6 26 Eample Write a logarithmic equation equivalent to Evaluate log , so log 8 6. Lesson 0-2 Eercises Write each equation in logarithmic form log log log Write each equation in eponential form.. log log log Evaluate each epression. 7. log 6 8. log log 00 00, log log log log 2 7. log log Glencoe/McGraw-Hill 579 Glencoe Algebra 2

11 0-2 DATE PERIOD Study Guide and Intervention (continued) Logarithms and Logarithmic Functions Solve Logarithmic Equations and Inequalities Logarithmic to If b, 0, and log b y, then b y. Eponential Inequality If b, 0, and log b y, then 0 b y. Property of Equality for If b is a positive number other than, Logarithmic Functions then log b log b y if and only if y. Property of Inequality for If b, then log b log b y if and only if y, Logarithmic Functions and log b log b y if and only if y. Eample Eample 2 Solve log 2 2. log 2 2 Original equation 2 2 Definition of logarithm 2 8 Simplify. Simplify. The solution is. Solve log 5 ( ). log 5 ( ) Original equation 0 5 Logarithmic to eponential inequality 25 Addition Property of Inequalities 2 Simplify. The solution set is 2. Eercises Solve each equation or inequality. 5. log log 2c 2 8. log log log (5 ) 2 6. log 8 ( 5) log ( ) log (2 ) 8. log 2 ( 2 6) log 2 (2 2) 9. log log 2 ( ). log log 8 ( ) 2 5. log log log 2 ( ) 5 6. log (2) log ( ) 2 8. log Glencoe/McGraw-Hill 580 Glencoe Algebra 2

12 0-2 DATE PERIOD Skills Practice Logarithms and Logarithmic Functions Write each equation in logarithmic form log log log 8 2. log Write each equation in eponential form. 5. log log log log Evaluate each epression. 9. log log 9. log log Lesson log. log 5 5. log log 27 Solve each equation or inequality. Check your solutions. 7. log log log y 0 0 y 20. log 2. log 2 n 2 n 22. log b log 6 ( 2) log 2 ( ) log ( 2) log () 26. log 6 (y 5) log 6 (2y ) y 8 Glencoe/McGraw-Hill 58 Glencoe Algebra 2

13 0-2 DATE PERIOD Practice (Average) Logarithms and Logarithmic Functions Write each equation in logarithmic form log log log log log log Write each equation in eponential form. 7. log log log 0. log log log Evaluate each epression.. log 8. log log 2 6. log log log 8 9. log log log 22. log 2. log 9 9 (n ) n 2. 2 log Solve each equation or inequality. Check your solutions. 25. log 0 n 26. log log log log 7 q 0 0 q 0. log 6 (2y 8) 2 y. log y 6 2. log n 2. log b log 8 ( 7) log 8 (7 ) 5. log 7 (8 20) log 7 ( 6) 6. log ( 2 2) log SOUND Sounds that reach levels of 0 decibels or more are painful to humans. What is the relative intensity of 0 decibels? 0 8. INVESTING Maria invests $000 in a savings account that pays 8% interest compounded annually. The value of the account A at the end of five years can be determined from the equation log A log[000( 0.08) 5 ]. Find the value of A to the nearest dollar. $69 Glencoe/McGraw-Hill 582 Glencoe Algebra 2

14 0-2 DATE PERIOD Reading to Learn Mathematics Logarithms and Logarithmic Functions Pre-Activity Why is a logarithmic scale used to measure sound? Read the introduction to Lesson 0-2 at the top of page 5 in your tetbook. How many 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 log 8. 8 b. Write a logarithmic equation that is equivalent to log c. Write an eponential equation that is equivalent to log 0. 0 d. Write a logarithmic equation that is equivalent to log e. What is the inverse of the function y 5? y log 5 f. What is the inverse of the function y log 0? y 0 2. Match each function with its graph. a. y IV b. y log I c. y II I. y II. y III. y Lesson 0-2 O O O. Indicate whether each of the following statements about the eponential function y log 5 is true or false. a. The y-ais is an asymptote 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 y-intercept is. false Helping You Remember. An important skill needed for working with logarithms is changing an equation between logarithmic and eponential forms. Using the words base, eponent, and logarithm, describe an easy way to remember and apply the part of the definition of logarithm that says, log b y if and only if b y. Sample answer: In these equations, b stands for base. In log form, b is the subscript, and in eponential form, b is the number that is raised to a power. A logarithm is an eponent, so y, which is the log in the first equation, becomes the eponent in the second equation. Glencoe/McGraw-Hill 58 Glencoe Algebra 2

15 0-2 Enrichment Musical Relationships The frequencies of notes in a musical scale that are one octave apart are related by an eponential equation. For the eight C notes on a piano, the equation is C n C 2 n, where C n represents the frequency of note C n.. Find the relationship between C and C Find the relationship between C and C. The frequencies of consecutive notes are related by a common ratio r. The general equation is f n f r n.. If the frequency of middle C is 26.6 cycles per second and the frequency of the net higher C is 52.2 cycles per second, find the common ratio r. (Hint: The two C s are 2 notes apart.) Write the answer as a radical epression.. Substitute decimal values for r and f to find a specific equation for f n. 5. Find the frequency of F # above middle C. 6. Frets are a series of ridges placed across the fingerboard of a guitar. They are spaced so that the sound made by pressing a string against one fret has about.0595 times the wavelength of the sound made by using the net fret. The general equation is w n w 0 (.0595) n. Describe the arrangement of the frets on a guitar. Glencoe/McGraw-Hill 58 Glencoe Algebra 2

16 0- Study Guide and Intervention Properties of Logarithms Properties of Logarithms Properties of eponents can be used to develop the following properties of logarithms. Product Property For all positive numbers m, n, and b, where b, of Logarithms log b mn log b m log b n. Quotient Property For all positive numbers m, n, and b, where b, of Logarithms log b m n log b m log b n. Power Property For any real number p and positive numbers m and b, of Logarithms where b, log b m p p log b m. Eample Use log 28.0 and log.269 to approimate the value of each epression. a. log 6 log 6 log ( 2 ) Eercises log 2 log 2 log b. log 7 28 log 7 log log 28 log c. log 256 log 256 log ( ) log (.269) Use log and log to evaluate each epression. 7. log log log log log log Lesson log log 2 6, log Use log and log to evaluate each epression. 0. log log log log log log log log log Glencoe/McGraw-Hill 585 Glencoe Algebra 2

17 0- Solve Logarithmic Equations You can use the properties of logarithms to solve equations involving logarithms. Eample Solve each equation. a. 2 log log log 25 2 log log log 25 Original equation log 2 log log 25 Power Property log log 25 Quotient Property 25 Property of Equality for Logarithmic Functions 2 00 Multiply each side by. 0 Take the square root of each side. Since logarithms are undefined for 0, 0 is an etraneous solution. The only solution is 0. b. log 2 log 2 ( 2) log 2 log 2 ( 2) Original equation log 2 ( 2) Product Property ( 2) 2 Definition of logarithm Distributive Property ( )( 2) 0 Factor. 2or Zero Product Property Subtract 8 from each side. Since logarithms are undefined for 0, is an etraneous solution. The only solution is 2. Eercises Study Guide and Intervention (continued) Properties of Logarithms 2 2 Solve each equation. Check your solutions.. log 5 log 5 2 log log 6 log 8 log log 6 25 log 6 log log 2 log 2 ( ) log log 6 2 log 6 log 6 ( ) 6. 2 log ( ) log ( ) 2 7. log 2 log log 2 0 2, log 2 2 log log (c ) log (c ) log 5 0. log 5 ( ) log 5 (2 ) Glencoe/McGraw-Hill 586 Glencoe Algebra 2

18 0- Skills Practice Properties of Logarithms Use log and log to approimate the value of each epression.. log log log log log log log log log log Solve each equation. Check your solutions.. log 0 27 log 0 2. log 7 2 log 7 b 8. log 5 log log log 6 2c log 6 8 log log 5 y log 5 8 log log 2 q log 2 log Lesson 0-7. log 9 2 log 9 5 log 9 w log 8 2 log 8 log 8 b 2 9. log 0 log 0 ( 5) log log log (2 ) log log d log log 0 y log 0 (2 y) 0 2. log 2 s 2 log log 2 ( ) log 2 ( ) log (n ) log (n 2) 26. log 5 0 log 5 2 log 5 2 log 5 a 5 Glencoe/McGraw-Hill 587 Glencoe Algebra 2

19 0- Practice (Average) Properties of Logarithms Use log and log to approimate the value of each epression.. log log log log log log log log Solve each equation. Check your solutions log 7 n log log 0 u log 0 8. log 6 log 6 9 log log 8 8 log 8 w log 8 2. log 9 (u ) log 9 5 log 9 2u 2. log 2 log 2 5 log log y log 6 log 6 6. log 2 d 5 log 2 2 log log 0 (m 5) log 0 m log log 0 (b ) log 0 b log 0 9. log 8 (t 0) log 8 (t ) log log (a ) log (a 2) log log 0 (r ) log 0 r log 0 (r ) log ( 2 ) log ( 2) log 2. log 0 log 0 w log 8 (n ) log 8 (n ) 25. log 5 ( 2 9) log 6 (9 5) log 6 ( 2 ) 27. log 6 (2 5) log 6 (7 0) log 2 (5y 2) log 2 ( 2y) log 0 (c 2 ) 2 log 0 (c ) 0 0. log 7 2 log 7 log 7 log SOUND The loudness L of a sound in decibels is given by L 0 log 0 R, where R is the sound s relative intensity. If the intensity of a certain sound is tripled, by how many decibels does the sound increase? about.8 db 2. EARTHQUAKES An earthquake rated at.5 on the Richter scale is felt by many people, and an earthquake rated at.5 may cause local damage. The Richter scale magnitude reading m is given by m log 0, where represents the amplitude of the seismic wave causing ground motion. How many times greater is the amplitude of an earthquake that measures.5 on the Richter scale than one that measures.5? 0 times Glencoe/McGraw-Hill 588 Glencoe Algebra 2

20 0- Reading to Learn Mathematics Properties of Logarithms Pre-Activity How are the properties of eponents and logarithms related? Reading the Lesson Read the introduction to Lesson 0- at the top of page 5 in your tetbook. Find the value of log Find the value of log 5 5. Find the value of log 5 (25 5). 2 Which of the following statements is true? B A. log 5 (25 5) (log 5 25) (log 5 5) B. log 5 (25 5) log 5 25 log 5 5. Each of the properties of logarithms can be stated in words or in symbols. Complete the statements of these properties in words. a. The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. b. The logarithm of a power is the product of the logarithm of the base and the eponent. c. The logarithm of a product is the sum of the logarithms of its factors. 2. State whether each of the following equations is true or false. If the statement is true, name the property of logarithms that is illustrated. a. log 0 log 0 log true; Quotient Property b. log 2 log log 8 false c. log log 2 9 true; Power Property d. log 8 0 log 8 5 log 8 6 false Lesson 0-. The algebraic process of solving the equation log 2 log 2 ( 2) leads to or 2. Does this mean that both and 2 are solutions of the logarithmic equation? Eplain your reasoning. Sample answer: No; 2 is a solution because it checks: log 2 2 log 2 (2 2) log 2 2 log 2 2. However, because log 2 ( ) and log 2 ( 2) are undefined, is an etraneous solution and must be eliminated. The only solution is 2. Helping You Remember. A good way to remember something is to relate it something you already know. Use words to eplain how the Product Property for eponents can help you remember the product property for logarithms. Sample answer: When you multiply two numbers or epressions with the same base, you add the eponents and keep the same base. Logarithms are eponents, so to find the logarithm of a product, you add the logarithms of the factors, keeping the same base. Glencoe/McGraw-Hill 589 Glencoe Algebra 2

21 0- DATE PERIOD Enrichment Spirals Consider an angle in standard position with its verte at a point O 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 by the polar coordinates (r, ), where r is the directed distance of the point from O and is the measure of the angle. Graphs in this system may be drawn on polar coordinate paper such as the kind shown below Use a calculator to complete the table for log 2 r. 2 0 (Hint: To find on a calculator, press 20 LOG r ) LOG 2 ).) r Plot the points found in Eercise on the grid above and connect to form a smooth curve. This type of spiral is called a logarithmic spiral because the angle measures are proportional to the logarithms of the radii. Glencoe/McGraw-Hill 590 Glencoe Algebra 2

22 0- Study Guide and Intervention Common Logarithms Common Logarithms Base 0 logarithms are called common logarithms. The epression log 0 is usually written without the subscript as log. Use the LOG key on your calculator to evaluate common logarithms. The relation between eponents and logarithms gives the following identity. Inverse Property of Logarithms and Eponents 0 log Eample Evaluate log 50 to four decimal places. Use the LOG key on your calculator. To four decimal places, log Eample 2 Solve Original equation log 2 log 2 (2 ) log log 2 Power Property of Logarithms log 2 2 log Divide each side by log. log 2 2 log Subtract from each side. Eercises log 2 2 log Property of Equality for Logarithms Multiply each side by. Use a calculator to evaluate each epression to four decimal places.. log 8 2. log 9. log log log log Solve each equation or inequality. Round to four decimal places { 0.859} Lesson {.5} Glencoe/McGraw-Hill 59 Glencoe Algebra 2

23 0- Study Guide and Intervention (continued) Common Logarithms Change of Base Formula The following formula is used to change epressions with different logarithmic bases to common logarithm epressions. Change of Base Formula For all positive numbers a, b, and n, where a and b, log a n log b n logb a Eample Epress log 8 5 in terms of common logarithms. Then approimate its value to four decimal places. log 0 5 log 8 5 Change of Base Formula log Simplify. The value of log 8 5 is approimately.02. Eercises Epress each logarithm in terms of common logarithms. Then approimate its value to four decimal places.. log 6 2. log 2 0. log 5 5 log 6 log 0 log 5, 2.527, 5.29, log log 2 log 5. log log log 2 50 log 22 log 200 log 50, , 2.22, 5.69 log log 2 log 2 7. log log 2 9. log 28.5 log 0. log 2 log 28.5, 0.569, 0.609, 2.6 log 5 log log 0. log (20) 2. log 6 (5) 2. log 8 () 5 2 log 20 log 5 5 log, 5.57,.590,. log log 6 log 8. log 5 (8). log 2 (.6) 6 5. log 2 (0.5) log 8 6 log.6 log 0.5,.876,.0880,.785 log 5 log 2 log 2 6. log log 9 8. log5 600 log 50 log 9 log 600, 2.280, ,.60 2 log log log 5 Glencoe/McGraw-Hill 592 Glencoe Algebra 2

24 0- Skills Practice Common Logarithms Use a calculator to evaluate each epression to four decimal places.. log log log log Use the formula ph log[h ] to find the ph of each substance given its concentration of hydrogen ions. 5. gastric juices: [H ].0 0 mole per liter.0 6. tomato juice: [H ] mole per liter. 7. blood: [H ] mole per liter toothpaste: [H ] mole per liter 9.9 Solve each equation or inequality. Round to four decimal places { 5} 0. 6 v v v 2. 8 p y b k p m r y 5 {y y } d t Lesson 0- Epress each logarithm in terms of common logarithms. Then approimate its value to four decimal places. log 2. log log ; log ; log0 log0 5 log log 25. log ; log ;.285 log0 2 log0 6 Glencoe/McGraw-Hill 59 Glencoe Algebra 2

25 0- Practice (Average) Common Logarithms Use a calculator to evaluate each epression to four decimal places.. log log log Use the formula ph log[h ] to find the ph of each substance given its concentration of hydrogen 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 inequality. Round to four decimal places {.69} 9. 5 a z m 00 {m m } y b 7. {b b.8699} a z 2 8 {z z.6555} 8. 5 w n 5 2n 0.97 Epress each logarithm in terms of common logarithms. Then approimate its value to four decimal places. log 2. log log ; log log ; log ; 0.96 log0 5 log0 8 log0 log log 26. log log ; log ; log ; 0.5 log0 2 log0 9 2 log HORTICULTURE Siberian irises flourish when the concentration of hydrogen 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 2.9 and the ph of milk is 6.6. How many times greater is the hydrogen ion concentration of vinegar than of milk? about BIOLOGY There are initially 000 bacteria in a culture. The number of bacteria doubles each hour. The number of bacteria N present after t hours is N 000(2) t. How long will it take the culture to increase to 50,000 bacteria? about 5.6 h 2. SOUND An equation for loudness L in decibels is given by L 0 log R, where R is the sound s relative intensity. An air-raid siren can reach 50 decibels and jet engine noise can reach 20 decibels. How many times greater is the relative intensity of the air-raid siren than that of the jet engine noise? 000 Glencoe/McGraw-Hill 59 Glencoe Algebra 2

26 0- DATE PERIOD Reading to Learn Mathematics Common Logarithms Pre-Activity Why is a logarithmic scale used to measure acidity? Reading the Lesson Read the introduction to Lesson 0- at the top of page 57 in your tetbook. Which substance is more acidic, milk or tomatoes? tomatoes. Rhonda used the following keystrokes to enter an epression on her graphing calculator: The calculator returned the result Which of the following conclusions are correct? a, c, and d a. The base 0 logarithm of 7 is about.20. b. The base 7 logarithm of 0 is about.20. c. The common logarithm of 7 is about.209. d is very close to 7. LOG e. The common logarithm of 7 is eactly Match each epression from the first column with an epression from the second column that has the same value. a. log 2 2 iv i. log b. log 2 iii ii. log 2 8 c. log i iii. log 0 2 d. log 5 v iv. log ) ENTER e. log 000 ii v. log 0.. Calculators do not have keys for finding base 8 logarithms directly. However, you can use a calculator to find log 8 20 if you apply the change of base formula. Which of the following epressions are equal to log 8 20? B and C log A. log 20 8 B log 20 log 8 C. D. log0 8 log 8 log 20 Lesson 0- Helping You Remember. Sometimes it is easier to remember a formula if you can state it in words. State the change of base formula in words. Sample answer: To change the logarithm of a number from one base to another, divide the log of the original number in the old base by the log of the new base in the old base. Glencoe/McGraw-Hill 595 Glencoe Algebra 2

27 0- Enrichment 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 logarithms. It has two movable rods labeled with C and D scales. Each of the scales is logarithmic. C D To multiply 2 on a slide rule, move the C rod to the right as shown below. You can find 2 by adding log 2 to log, and the slide rule adds the lengths for you. The distance you get is 0.778, or the logarithm of 6. log 2 log D C log 6 Follow the steps to make a slide rule.. Use graph paper that has small squares, such as 0 squares to the inch. Using the scales shown at the right, plot the curve y log for,.5, and the whole numbers from 2 through 0. Make an obvious heavy dot for each point plotted You will need two strips of cardboard. A 5-by-7 inde card, cut in half the long way, will work fine. Turn the graph you made in Eercise sideways and use it to mark a logarithmic scale on each of the two strips. The figure shows the mark for 2 being drawn y. Eplain how to use a slide rule to divide 8 by 2. 2 y = log.5 2 Glencoe/McGraw-Hill 596 Glencoe Algebra 2

28 0-5 Study Guide and Intervention Base e and Natural Logarithms Base e and Natural Logarithms The irrational number e often occurs as the base for eponential and logarithmic functions that describe real-world phenomena. Natural Base e As n increases, n approaches e n ln log e The functions y e and y ln are inverse functions. Inverse Property of Base e and Natural Logarithms e ln ln e Natural base epressions can be evaluated using the e and ln keys on your calculator. Eample Evaluate ln 685. Use a calculator. ln Eample 2 Write a logarithmic equation equivalent to e 2 7. e 2 7 log e 7 2 or 2 ln 7 Eample Evaluate ln e 8. Use the Inverse Property of Base e and Natural Logarithms. ln e 8 8 Eercises Use a calculator to evaluate each epression to four decimal places.. ln ln 8,50. ln ln ln ln ln 28,25 8. ln Write an equivalent eponential or logarithmic equation. 9. e 5 0. e 5. ln ln 8 ln 5 ln 5 e 20 e 8. e ln () e ln ln 0.2 e 9.6 ln e Evaluate each epression. 7. ln e 8. e ln 2 9. e ln ln e Lesson 0-5 Glencoe/McGraw-Hill 597 Glencoe Algebra 2

29 0-5 Equations and Inequalities with e and ln All properties of logarithms from earlier lessons can be used to solve equations and inequalities with natural logarithms. Eample a. e e Solve each equation or inequality. Original equation e 2 8 e 2 8 Divide each side by. ln e 2 ln 0.90 b. ln ( ) 2 ln ( ) 2 Study Guide and Intervention (continued) Base e and Natural Logarithms 8 Subtract 2 from each side. Property of Equality for Logarithms 8 2 ln 8 ln Multiply each side by 2 2. Inverse Property of Eponents and Logarithms Use a calculator. Original inequality e ln ( ) e 2 Write each side using eponents and base e. 0 e 2 Inverse Property of Eponents and Logarithms e 2 Addition Property of Inequalities (e 2 ) Multiplication Property of Inequalities Use a calculator. Eercises Solve each equation or inequality.. e e 25. e {.289}.82. ln 6 5. ln ( ) e { 0.890} 7. e 2 8. ln (5 ) e no solution 0. 6 e 2. ln (2 5) 8 2. ln 5 ln { 2.22} Glencoe/McGraw-Hill 598 Glencoe Algebra 2

30 0-5 Skills Practice Base e and Natural Logarithms Use a calculator to evaluate each epression to four decimal places.. e e ln ln Write an equivalent eponential or logarithmic equation. 5. e ln 6. e 8 ln 8 7. ln 5 e 5 8. ln 0.69 e 0.69 Evaluate each epression. 9. e ln 0. e ln 2 2. ln e ln e y y Solve each equation or inequality.. e 5 {.609}. e.2 {.62} 5. 2e e e e 0 { } 9. e 5 { } 20. 2e ln ln ln ( 2) ln ( ) ln ( ) ln ln Lesson 0-5 Glencoe/McGraw-Hill 599 Glencoe Algebra 2

31 0-5 Practice (Average) Base e and Natural Logarithms Use a calculator to evaluate each epression to four decimal places.. e ln ln e e ln 0 7. e ln Write an equivalent eponential or logarithmic equation. 9. ln ln 6 2. ln ln e 50 e 2 6 e e e 8. e e 6. e 2 ln 8 5 ln 0 ln 2 ln ( ) Evaluate each epression. 7. e ln e ln 9. ln e 20. ln e 2y 2y Solve each equation or inequality. 2. e e 2. e. 2. e 5.8 { 2.972} e 26. 5e e e { 0.82} { 0.055} 29. e 8 0. e 5. e e e e e e { 0.69} 7. ln 8. ln ( 2) 7 9. ln ln ( 6) ln ( 2) 2. ln ( ) 5. ln ln 2 9. ln 5 ln INVESTING For Eercises 5 and 6, use the formula for continuously compounded interest, A Pe rt, where P is the principal, r is the annual interest rate, and t is the time in years. 5. If Sarita deposits $000 in an account paying.% annual interest compounded continuously, what is the balance in the account after 5 years? $ How long will it take the balance in Sarita s account to reach $2000? about 20. yr 7. RADIOACTIVE DECAY The amount of a radioactive substance y that remains after t years is given by the equation y ae kt, where a is the initial amount present and k is the decay constant for the radioactive substance. If a 00, y 50, and k 0.05, find t. about 9.8 yr Glencoe/McGraw-Hill 600 Glencoe Algebra 2

32 0-5 DATE PERIOD Reading to Learn Mathematics Base e and Natural Logarithms Pre-Activity How is the natural base e used in banking? Reading the Lesson Read the introduction to Lesson 0-5 at the top of page 55 in your tetbook. Suppose that you deposit $675 in a savings account that pays an annual interest rate of 5%. In each case listed below, indicate which method of compounding would result in more money in your account at the end of one year. a. annual compounding or monthly compounding monthly b. quarterly compounding or daily compounding daily c. daily compounding or continuous compounding continuous. Jagdish entered the following keystrokes in his calculator: LN 5 The calculator returned the result Which of the following conclusions are correct? d and f a. The common logarithm of 5 is about.609. b. The natural logarithm of 5 is eactly c. The base 5 logarithm of e is about.609. d. The natural logarithm of 5 is about e is very close to 5. f. e is very close to Match each epression from the first column with its value in the second column. Some choices may 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 e III VI. e ) ENTER Helping You Remember. A good way to remember something is to eplain it to someone else. Suppose that you are studying with a classmate who is puzzled when asked to evaluate ln e.how would you eplain to him an easy way to figure this out? Sample answer: ln means natural log.the natural log of e is the power to which you raise e to get e.this is obviously. Lesson 0-5 Glencoe/McGraw-Hill 60 Glencoe Algebra 2

33 0-5 Enrichment 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 closely. n n Another way 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 by the English mathematician John Wallis (66 70) n 2n 2n 2n Solve each problem.. Use a calculator with an e key to find e to 7 decimal places. 2. Use the epression n n to approimate e to decimal places. Use 5, 00, 500, and 7000 as values of n.. Use the infinite sum to approimate e to decimal places. Use the whole numbers from through 6 as values of n.. Which approimation method approaches the value of e more quickly? 5. Use a calculator with a key to find to 7 decimal places. 6. Use the infinite product to approimate to decimal places. Use the whole numbers from through 6 as values of n. 7. Does the infinite product give good approimations for quickly? 8. Show that 5 is equal to e 6 to decimal places. 9. Which is larger, e or e? 0. The epression reaches a maimum value at e. Use this fact to prove the inequality you found in Eercise 9. Glencoe/McGraw-Hill 602 Glencoe Algebra 2

34 0-6 Study Guide and Intervention Eponential Growth and Decay Eponential Decay Depreciation of value and radioactive decay are eamples of eponential decay. When a quantity decreases by a fied percent each time period, the amount of the quantity after t time periods is given by y a( r) t, where a is the initial amount and r is the percent decrease epressed as a decimal. Another eponential decay model often used by scientists is y ae kt, where k is a constant. Eample CONSUMER PRICES As technology advances, the price of many technological devices such as scientific calculators and camcorders goes down. One brand of hand-held organizer sells for $89. a. If its price decreases by 6% per year, how much will it cost after 5 years? Use the eponential decay model with initial amount $89, percent decrease 0.06, and time 5 years. y a( r) t Eponential decay formula y 89( 0.06) 5 a 89, r 0.06, t 5 y $65.2 After 5 years the price will be $65.2. b. After how many years will its price be $50? To find when the price will be $50, again use the eponential decay formula and solve for t. y a( r) t Eponential decay formula 50 89( 0.06) t y 50, a 89, r (0.9) t Divide each side by 89. Lesson log log (0.9) t Property of Equality for Logarithms log t log 0.9 Power Property log t Divide each side by log 0.9. log 0.9 t 9. The price will be $50 after about 9. years. Eercises. BUSINESS A furniture store is closing out its business. Each week the owner lowers prices by 25%. After how many weeks will the sale price of a $500 item drop below $00? 6 weeks CARBON DATING Use the formula y ae t, where a is the initial amount of Carbon-, t is the number of years ago the animal lived, and y is the remaining amount after t years. 2. How old is a fossil remain that has lost 95% of its Carbon-? about 25,000 years old. How old is a skeleton that has 95% of its Carbon- remaining? about 27.5 years old Glencoe/McGraw-Hill 60 Glencoe Algebra 2

35 0-6 Eponential Growth Population increase and growth of bacteria colonies are eamples of eponential growth. When a quantity increases by a fied percent each time period, the amount of that quantity after t time periods is given by y a( r) t, where a is the initial amount and r is the percent increase (or rate of growth) epressed as a decimal. Another eponential growth model often used by scientists is y ae kt, where k is a constant. Eample Study Guide and Intervention (continued) Eponential Growth and Decay A computer engineer is hired for a salary of $28,000. If she gets a 5% raise each year, after how many years will she be making $50,000 or more? Use the eponential growth model with a 28,000, y 50,000, and r 0.05 and solve for t. y a( r) t Eponential growth formula 50,000 28,000( 0.05) t y 50,000, a 28,000, r (.05) t Divide each side by 28, log log (.05) t Property of Equality of Logarithms log t log.05 Power Property log t log.05 Divide each side by log.05. t.9 years Use a calculator. If raises are given annually, she will be making over $50,000 in 2 years. Eercises. BACTERIA GROWTH A certain strain of bacteria grows from 0 to 26 in 20 minutes. Find k for the growth formula y ae kt, where t is in minutes. about INVESTMENT Carl plans to invest $500 at 8.25% interest, compounded continuously. How long will it take for his money to triple? about years. SCHOOL POPULATION There are currently 850 students at the high school, which represents full capacity. The town plans an addition to house 00 more students. If the school population grows at 7.8% per year, in how many years will the new addition be full? about 5 years. EXERCISE Hugo begins a walking program by walking mile per day for one week. 2 Each week thereafter he increases his mileage by 0%. After how many weeks is he walking more than 5 miles per day? 2 weeks 5. VOCABULARY GROWTH When Emily was 8 months old, she had a 0-word vocabulary. By the time she was 5 years old (60 months), her vocabulary was 2500 words. If her vocabulary increased at a constant percent per month, what was that increase? about % Glencoe/McGraw-Hill 60 Glencoe Algebra 2

36 0-6 Skills Practice Eponential Growth and Decay Solve each problem.. FISHING In an over-fished area, the catch of a certain fish is decreasing at an average rate of 8% per year. If this decline persists, how long will it take for the catch to reach half of the amount before the decline? about 8. yr Lesson INVESTING Ale invests $2000 in an account that has a 6% annual rate of growth. To the nearest year, when will the investment be worth $600? 0 yr. POPULATION A current census shows that the population of a city is.5 million. Using the formula P ae rt, find the epected population of the city in 0 years if the growth rate r of the population is.5% per year, a represents the current population in millions, and t represents the time in years. about 5.5 million. POPULATION The population P in thousands of a city can be modeled by the equation P 80e 0.05t, where t is the time in years. In how many years will the population of the city be 20,000? about 27 yr 5. BACTERIA How many days will it take a culture of bacteria to increase from 2000 to 50,000 if the growth rate per day is 9.2%? about.9 days 6. NUCLEAR POWER The element plutonium-29 is highly radioactive. Nuclear reactors can produce and also use this element. The heat that plutonium-29 emits has helped to power equipment on the moon. If the half-life of plutonium-29 is 2,60 years, what is the value of k for this element? about DEPRECIATION A Global Positioning Satellite (GPS) system uses satellite information to locate ground position. Abu s surveying firm bought a GPS system for $2,500. The GPS depreciated by a fied rate of 6% and is now worth $8600. How long ago did Abu buy the GPS system? about 6.0 yr 8. BIOLOGY In a laboratory, an organism grows from 00 to 250 in 8 hours. What is the hourly growth rate in the growth formula y a( r) t? about 2.% Glencoe/McGraw-Hill 605 Glencoe Algebra 2

37 0-6 Solve each problem. Practice (Average) Eponential Growth and Decay r 2. INVESTING The formula A P 2t gives the value of an investment after t years in an account that earns an annual interest rate r compounded twice a year. Suppose $500 is invested at 6% annual interest compounded twice a year. In how many years will the investment be worth $000? about.7 yr 2. BACTERIA How many hours will it take a culture of bacteria to increase from 20 to 2000 if the growth rate per hour is 85%? about 7.5 h. RADIOACTIVE DECAY A radioactive substance has a half-life of 2 years. Find the constant k in the decay formula for the substance. about DEPRECIATION A piece of machinery valued at $250,000 depreciates at a fied rate of 2% per year. After how many years will the value have depreciated to $00,000? about 7.2 yr 5. INFLATION For Dave to buy a new car comparably equipped to the one he bought 8 years ago would cost $2,500. Since Dave bought the car, the inflation rate for cars like his has been at an average annual rate of 5.%. If Dave originally paid $800 for the car, how long ago did he buy it? about 8 yr 6. RADIOACTIVE DECAY Cobalt, an element used to make alloys, has several isotopes. One of these, cobalt-60, is radioactive and has a half-life of 5.7 years. Cobalt-60 is used to trace the path of nonradioactive substances in a system. What is the value of k for Cobalt-60? about WHALES Modern whales appeared 5 0 million years ago. The vertebrae of a whale discovered by paleontologists contain roughly 0.25% as much carbon- as they would have contained when the whale was alive. How long ago did the whale die? Use k about 50,000 yr 8. POPULATION The population of rabbits in an area is modeled by the growth equation P(t) 8e 0.26t, where P is in thousands and t is in years. How long will it take for the population to reach 25,000? about. yr 9. DEPRECIATION A computer system depreciates at an average rate of % per month. If the value of the computer system was originally $2,000, in how many months is it worth $750? about 2 mo 0. BIOLOGY In a laboratory, a culture increases from 0 to 95 organisms in 5 hours. What is the hourly growth rate in the growth formula y a( r) t? about 5.% Glencoe/McGraw-Hill 606 Glencoe Algebra 2

38 0-6 Reading to Learn Mathematics Eponential Growth and Decay Pre-Activity How can you determine the current value of your car? Read the introduction to Lesson 0-6 at the top of page 560 in your tetbook. Between which two years shown in the table did the car depreciate by the greatest amount? between years 0 and Describe two ways to calculate the value of the car 6 years after it was purchased. (Do not actually calculate the value.) Sample answer:. Multiply $ by 0.6 and subtract the result from $ Multiply $ by 0.8. Lesson 0-6 Reading the Lesson. State whether each situation is an eample of eponential growth or decay. a. A city had 2,000 residents in 980 and 28,000 residents in growth b. Raul compared the value of his car when he bought it new to the value when he traded ;lpit in si years later. decay c. A paleontologist compared the amount of carbon- in the skeleton of an animal when it died to the amount 00 years later. decay d. Maria deposited $750 in a savings account paying.5% annual interest compounded quarterly. She did not make any withdrawals or further deposits. She compared the balance in her passbook immediately after she opened the account to the balance years later. growth 2. State whether each equation represents eponential growth or decay. a. y 5e 0.5t growth b. y 000( 0.05) t decay c. y 0.e 200t decay d. y 2( 0.000) t growth Helping You Remember. Visualizing their graphs is often a good way to remember the difference between mathematical equations. How can your knowledge of the graphs of eponential equations from Lesson 0- help you to remember that equations of the form y a( r) t represent eponential growth, while equations of the form y a( r) t represent eponential decay? Sample answer: If a 0, the graph of y ab is always increasing if b and is always decreasing if 0 b. Since r is always 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 (decay). Glencoe/McGraw-Hill 607 Glencoe Algebra 2