7 Eponential and Logarithmic Functions 7.1 Eponential Growth and Deca Functions 7. The Natural Base e 7.3 Logarithms and Logarithmic Functions 7. Transformations of Eponential and Logarithmic Functions 7.5 Properties of Logarithms 7. Solving Eponential and Logarithmic Equations 7.7 Modeling with Eponential and Logarithmic Functions SEE the Big Idea Astronaut Health (p. 399) Cooking (p. 387) Recording Studio (p. 38) Tornado Wind Speed (p. 37) Duckweed Growth (p. 353) Mathematical Thinking: Mathematicall proficient students can appl the mathematics the know to solve problems arising in everda life, societ, and the workplace.
Maintaining Mathematical Proficienc Using Eponents (.7.A) Eample 1 Evaluate ( 3) 1. ( 1 3) = ( 3) 1 ( 1 3) ( 1 3) ( 1 3) Rewrite ( 1 = ( 9) 1 ( 3) 1 ( 1 3) Multipl. = ( 1 = 1 81 Evaluate the epression. 7) ( 1 3) Multipl. 1. 3. ( ) 5 3. ( ) 5. ( ) 3 3 Multipl. Finding the Domain and Range of a Function (A..A) Eample Find the domain and range of the function represented b the graph. 3) as repeated multiplication. 3 3 1 1 range 3 domain The domain is { 3 3}. The range is { 1}. Find the domain and range of the function represented b the graph. 5.. 7. 8. ABSTRACT REASONING Consider the epressions n and ( ) n, where n is an integer. For what values of n is each epression negative? positive? Eplain our reasoning. 35
Mathematical Thinking Selecting Tools Core Concept Using a Spreadsheet To use a spreadsheet, it is common to write one cell as a function of another cell. For instance, in the spreadsheet shown, the cells in column A starting with cell A contain functions of the cell in the preceding row. Also, the cells in column B contain functions of the cells in the same row in column A. Mathematicall profi cient students select tools, including real objects, manipulatives, paper and pencil, and technolog as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. (A.1.C) A = A1+1 1 3 5 A 1 3 5 7 7 8 8 B 0 8 10 1 1 B1 = *A1 Using a Spreadsheet You deposit $1000 in stocks that earn 15% interest compounded annuall. Use a spreadsheet to find the balance at the end of each ear for 8 ears. Describe the tpe of growth. You can enter the given information into a spreadsheet and generate the graph shown. From the formula in the spreadsheet, ou can see that the growth pattern is eponential. The graph also appears to be eponential. 1 3 5 7 8 9 10 1 A Year 0 1 3 5 7 8 B Balance $1000.00 $1150.00 $13.50 $150.88 $179.01 $011.3 $313.0 $0.0 $3059.0 Monitoring Progress B3 = B*1.15 Use a spreadsheet to help ou answer the question. 1. A pilot flies a plane at a speed of 500 miles per hour for hours. Find the total distance flown at 30 minute intervals. Describe the pattern.. A population of 0 rabbits increases b 5% each ear for 8 ears. Find the population at the end of each ear. Describe the tpe of growth. 3. An endangered population has 500 members. The population declines b 10% each decade for 80 ears. Find the population at the end of each decade. Describe the tpe of decline.. The top eight runners finishing a race receive cash prizes. First place receives $00, second place receives $175, third place receives $150, and so on. Find the fifth through eighth place prizes. Describe the tpe of decline. Balance (dollars) $3500.00 $3000.00 $500.00 $000.00 $1500.00 $1000.00 0 Stock Investment 8 10 Year 3 Chapter 7 Eponential and Logarithmic Functions
7.1 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A Eponential Growth and Deca Functions Essential Question What are some of the characteristics of the graph of an eponential function? You can use a graphing calculator to evaluate an eponential function. For eample, consider the eponential function f () =. Function Value Graphing Calculator Kestrokes Displa f ( 3.1) = 3.1 3.1 ENTER 0.1191 f ( 3 ) = /3 ( 3 ) ENTER 1.587011 Identifing Graphs of Eponential Functions Work with a partner. Match each eponential function with its graph. Use a table of values to sketch the graph of the function, if necessar. a. f () = b. f () = 3 c. f () = d. f () = ( 1 ) e. f () = ( 1 3 ) f. f () = ( 1 ) A. B. C. D. E. F. MAKING MATHEMATICAL ARGUMENTS To be proficient in math, ou need to justif our conclusions and communicate them to others. Characteristics of Graphs of Eponential Functions Work with a partner. Use the graphs in Eploration 1 to determine the domain, range, and -intercept of the graph of f () = b, where b is a positive real number other than 1. Eplain our reasoning. Communicate Your Answer 3. What are some of the characteristics of the graph of an eponential function?. In Eploration, is it possible for the graph of f () = b to have an -intercept? Eplain our reasoning. Section 7.1 Eponential Growth and Deca Functions 37
7.1 Lesson What You Will Learn Core Vocabular eponential function, p. 38 eponential growth function, p. 38 growth factor, p. 38 asmptote, p. 38 eponential deca function, p. 38 deca factor, p. 38 Previous properties of eponents Graph eponential growth and deca functions. Use eponential models to solve real-life problems. Eponential Growth and Deca Functions An eponential function has the form = ab, where a 0 and the base b is a positive real number other than 1. If a > 0 and b > 1, then = ab is an eponential growth function, and b is called the growth factor. The simplest tpe of eponential growth function has the form = b. Core Concept Parent Function for Eponential Growth Functions The function f () = b, where b > 1, is the parent function for the famil of eponential growth functions with base b. The graph shows the general shape of an eponential growth function. The -ais is an asmptote of the graph. An asmptote is a line that a graph approaches more and more closel. (0, 1) f() = b (b > 1) The graph rises from left to right, passing (1, b) through the points (0, 1) and (1, b). The domain of f () = b is all real numbers. The range is > 0. If a > 0 and 0 < b < 1, then = ab is an eponential deca function, and b is called the deca factor. Core Concept Parent Function for Eponential Deca Functions The function f () = b, where 0 < b < 1, is the parent function for the famil of eponential deca functions with base b. The graph shows the general shape of an eponential deca function. The graph falls from left to right, passing through the points (0, 1) and (1, b). f() = b (0 < b < 1) (0, 1) (1, b) The -ais is an asmptote of the graph. The domain of f () = b is all real numbers. The range is > 0. 38 Chapter 7 Eponential and Logarithmic Functions
Graphing Eponential Growth and Deca Functions Tell whether each function represents eponential growth or eponential deca. Then graph the function. a. = b. = ( 1 ) a. Step 1 Identif the value of the base. The base,, is greater than 1, so the function represents eponential growth. Step Make a table of values. 1 0 1 3 8 (3, 8) 1 1 1 8 = Step 3 Plot the points from the table. 1 (, ) ( 1, Step Draw, from left to right, a smooth curve that 1 (1, ) begins just above the -ais, passes through (, (0, 1) the plotted points, and moves up to the right. b. Step 1 Identif the value of the base. The base, 1, is greater than 0 and less than 1, so the function represents eponential deca. Step Make a table of values. ( ( 3 1 0 1 8 1 1 1 Step 3 Plot the points from the table. Step Draw, from right to left, a smooth curve that begins just above the -ais, passes through the plotted points, and moves up to the left. ( 3, 8) (, ) 8 ( 1, ) (0, 1) 1 = ( 1 (1, 1 (, ( ( ( Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the function represents eponential growth or eponential deca. Then graph the function. 1. =. = ( 3 ) 3. f () = (0.5). f () = (1.5) Eponential Models Some real-life quantities increase or decrease b a fied percent each ear (or some other time period). The amount of such a quantit after t ears can be modeled b one of these equations. Eponential Growth Model = a(1 + r) t Eponential Deca Model = a(1 r) t Note that a is the initial amount and r is the percent increase or decrease written as a decimal. The quantit 1 + r is the growth factor, and 1 r is the deca factor. Section 7.1 Eponential Growth and Deca Functions 39
Solving a Real-Life Problem REASONING The percent decrease, 15%, tells ou how much value the car loses each ear. The deca factor, 0.85, tells ou what fraction of the car s value remains each ear. The value of a car (in thousands of dollars) can be approimated b the model = 5(0.85) t, where t is the number of ears since the car was new. a. Tell whether the model represents eponential growth or eponential deca. b. Identif the annual percent increase or decrease in the value of the car. c. Estimate when the value of the car will be $8000. a. The base, 0.85, is greater than 0 and less than 1, so the model represents eponential deca. b. Because t is given in ears and the deca factor 0.85 = 1 0.15, the annual percent decrease is 0.15, or 15%. 30 c. Use the trace feature of a graphing calculator to determine that 8 when t = 7. After 7 ears, = 5(0.85) the value of the car will be about $8000. 1 X=7 Y=8.017 5 15 X 7 8 9 10 11 1 X=1 Y1.531.11.89.781.88.988 7.010 Writing an Eponential Model In 000, the world population was about.09 billion. During the net 13 ears, the world population increased b about 1.18% each ear. a. Write an eponential growth model giving the population (in billions) t ears after 000. Estimate the world population in 005. b. Estimate the ear when the world population was 7 billion. a. The initial amount is a =.09, and the percent increase is r = 0.0118. So, the eponential growth model is = a(1 + r) t Write eponential growth model. =.09(1 + 0.0118) t Substitute.09 for a and 0.0118 for r. =.09(1.0118) t. Simplif. Using this model, ou can estimate the world population in 005 (t = 5) to be =.09(1.0118) 5. billion. b. Use the table feature of a graphing calculator to determine that 7 when t = 1. So, the world population was about 7 billion in 01. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. WHAT IF? In Eample, the value of the car can be approimated b the model = 5(0.9) t. Identif the annual percent decrease in the value of the car. Estimate when the value of the car will be $8000.. WHAT IF? In Eample 3, assume the world population increased b 1.5% each ear. Write an equation to model this situation. Estimate the ear when the world population was 7 billion. 350 Chapter 7 Eponential and Logarithmic Functions
Rewriting an Eponential Function The amount (in grams) of the radioactive isotope chromium-51 remaining after t das is = a(0.5) t/8, where a is the initial amount (in grams). What percent of the chromium-51 decas each da? = a(0.5) t/8 Write original function. = a[(0.5) 1/8 ] t Power of a Power Propert a(0.9755) t Evaluate power. = a(1 0.05) t Rewrite in form = a(1 r) t. The dail deca rate is about 0.05, or.5%. Compound interest is interest paid on an initial investment, called the principal, and on previousl earned interest. Interest earned is often epressed as an annual percent, but the interest is usuall compounded more than once per ear. So, the eponential growth model = a(1 + r) t must be modified for compound interest problems. Core Concept Compound Interest Consider an initial principal P deposited in an account that pas interest at an annual rate r (epressed as a decimal), compounded n times per ear. The amount A in the account after t ears is given b A = P ( 1 + r n ) nt. Finding the Balance in an Account You deposit $9000 in an account that pas 1.% annual interest. Find the balance after 3 ears when the interest is compounded quarterl. With interest compounded quarterl ( times per ear), the balance after 3 ears is A = P ( 1 + r n ) nt = 9000 ( 1 + 0.01 ) 3 90.1. Write compound interest formula. P = 9000, r = 0.01, n =, t = 3 Use a calculator. The balance at the end of 3 ears is $90.1. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 7. The amount (in grams) of the radioactive isotope iodine-13 remaining after t hours is = a(0.5) t/13, where a is the initial amount (in grams). What percent of the iodine-13 decas each hour? 8. WHAT IF? In Eample 5, find the balance after 3 ears when the interest is compounded dail. Section 7.1 Eponential Growth and Deca Functions 351
7.1 Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY In the eponential growth model =.(1.5), identif the initial amount, the growth factor, and the percent increase.. WHICH ONE DOESN T BELONG? Which characteristic of an eponential deca function does not belong with the other three? Eplain our reasoning. base of 0.8 deca factor of 0.8 deca rate of 0% 80% decrease Monitoring Progress and Modeling with Mathematics In Eercises 3 8, evaluate the epression for (a) = and (b) = 3. 3.. 5. 8 3. 7. 5 + 3 8. In Eercises 9 18, tell whether the function represents eponential growth or eponential deca. Then graph the function. (See Eample 1.) 9. = 10. = 7 11. = ( 1 ) 13. = ( 3) 1. = ( 8) 1 1. = ( 5) 15. = (1.) 1. = (0.75) 17. = (0.) 18. = (1.8) ANALYZING RELATIONSHIPS In Eercises 19 and 0, use the graph of f() = b to identif the value of the base b. 19. 1 ( 1, 3 ( (1, 3) (0, 1) 0. 1 ( 1, 5 ( (1, 5) (0, 1) 1. MODELING WITH MATHEMATICS The value of a mountain bike (in dollars) can be approimated b the model = 00(0.75) t, where t is the number of ears since the bike was new. (See Eample.) a. Tell whether the model represents eponential growth or eponential deca. b. Identif the annual percent increase or decrease in the value of the bike. c. Estimate when the value of the bike will be $50.. MODELING WITH MATHEMATICS The population P (in thousands) of Austin, Teas, during a recent decade can be approimated b = 9.9(1.03) t, where t is the number of ears since the beginning of the decade. a. Tell whether the model represents eponential growth or eponential deca. b. Identif the annual percent increase or decrease in population. c. Estimate when the population was about 590,000. 3. MODELING WITH MATHEMATICS In 00, there were approimatel 33 million cell phone subscribers in the United States. During the net ears, the number of cell phone subscribers increased b about % each ear. (See Eample 3.) a. Write an eponential growth model giving the number of cell phone subscribers (in millions) t ears after 00. Estimate the number of cell phone subscribers in 008. b. Estimate the ear when the number of cell phone subscribers was 75 million. 35 Chapter 7 Eponential and Logarithmic Functions
. MODELING WITH MATHEMATICS You take a 35 milligram dosage of ibuprofen. During each subsequent hour, the amount of medication in our bloodstream decreases b about 9% each hour. a. Write an eponential deca model giving the amount (in milligrams) of ibuprofen in our bloodstream t hours after the initial dose. b. Estimate how long it takes for ou to have 100 milligrams of ibuprofen in our bloodstream. JUSTIFYING STEPS In Eercises 5 and, justif each step in rewriting the eponential function. 5. = a(3) t/1 Write original function. = a[(3) 1/1 ] t = a(1.081) t = a(1 + 0.081) t. = a(0.1) t/3 Write original function. = a[(0.1) 1/3 ] t = a(0.) t = a(1 0.5358) t 7. PROBLEM SOLVING When a plant or animal dies, it stops acquiring carbon-1 from the atmosphere. The amount (in grams) of carbon-1 in the bod of an organism after t ears is = a(0.5) t/5730, where a is the initial amount (in grams). What percent of the carbon-1 is released each ear? (See Eample.) 8. PROBLEM SOLVING The number of duckweed fronds in a pond after t das is = a(130.5) t/1, where a is the initial number of fronds. B what percent does the duckweed increase each da? 33. = a( 3 ) t/10 3. = a( 5 ) t/ 35. = a() 8t 3. = a( 1 3 ) 3t 37. PROBLEM SOLVING You deposit $5000 in an account that pas.5% annual interest. Find the balance after 5 ears when the interest is compounded quarterl. (See Eample 5.) 38. DRAWING CONCLUSIONS You deposit $00 into three separate bank accounts that each pa 3% annual interest. How much interest does each account earn after ears? Account Compounding 1 quarterl monthl 3 dail Balance after ears 39. ERROR ANALYSIS You invest $500 in the stock of a compan. The value of the stock decreases % each ear. Describe and correct the error in writing a model for the value of the stock after t ears. = ( amount) Initial ( factor) Deca t = 500(0.0) t 0. ERROR ANALYSIS You deposit $50 in an account that pas 1.5% annual interest. Describe and correct the error in finding the balance after 3 ears when the interest is compounded quarterl. A = 50 ( 1 + 1.5 ) 3 A = $533.9 In Eercises 9 3, rewrite the function in the form = a(1 + r) t or = a(1 r) t. Then state the growth or deca rate. 9. = a() t/3 30. = a() t/ 31. = a(0.5) t/1 3. = a(0.5) t/9 In Eercises 1, use the given information to find the amount A in the account earning compound interest after ears when the principal is $3500. 1. r =.1%, compounded quarterl. r =.9%, compounded monthl 3. r = 1.83%, compounded dail. r = 1.%, compounded monthl Section 7.1 Eponential Growth and Deca Functions 353
5. USING STRUCTURE A website recorded the number of referrals it received from social media websites over a 10-ear period. The results can be modeled b = 500(1.50) t, where t is the ear and 0 t 9. Interpret the values of a and b in this situation. What is the annual percent increase? Eplain.. HOW DO YOU SEE IT? Consider the graph of an eponential function of the form f () = ab. ( 1, ) (0, 1) (1, 1 (, ( 1 1 ( 50. REASONING Consider the eponential function f () = ab. f ( + 1) a. Show that = b. f () b. Use the equation in part (a) to eplain wh there is no eponential function of the form f () = ab whose graph passes through the points in the table below. 0 1 3 8 7 51. PROBLEM SOLVING The number E of eggs a Leghorn chicken produces per ear can be modeled b the equation E = 179.(0.89) w/5, where w is the age (in weeks) of the chicken and w. a. Determine whether the graph of f represents eponential growth or eponential deca. b. What are the domain and range of the function? Eplain. 7. MAKING AN ARGUMENT Your friend sas the graph of f () = increases at a faster rate than the graph of g () = when 0. Is our friend correct? Eplain our reasoning. 8 g 0 0 8. THOUGHT PROVOKING The function f () = b represents an eponential deca function. Write a second eponential deca function in terms of b and. 9. PROBLEM SOLVING The population p of a small town after ears can be modeled b the function p = 850(1.03). What is the average rate of change in the population over the first ears? Justif our answer. Maintaining Mathematical Proficienc Simplif the epression. Assume all variables are positive. (Skills Review Handbook) 53. 9 5. 57. + 3 58. a. Identif the deca factor and the percent decrease. b. Graph the model. c. Estimate the egg production of a chicken that is.5 ears old. d. Eplain how ou can rewrite the given equation so that time is measured in ears rather than in weeks. 5. CRITICAL THINKING You bu a new stereo for $1300 and are able to sell it ears later for $75. Assume that the resale value of the stereo decas eponentiall with time. Write an equation giving the resale value V (in dollars) of the stereo as a function of the time t (in ears) since ou bought it. Reviewing what ou learned in previous grades and lessons 3 55. ( 5. 8 ) + 59. 1 + 5 0. ( 35 ) 3 35 Chapter 7 Eponential and Logarithmic Functions
7. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A SELECTING TOOLS To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. The Natural Base e Essential Question What is the natural base e? So far in our stud of mathematics, ou have worked with special numbers such as π and i. Another special number is called the natural base and is denoted b e. The natural base e is irrational, so ou cannot find its eact value. Approimating the Natural Base e Work with a partner. One wa to approimate the natural base e is to approimate the sum 1 + 1 1 + 1 1 + 1 1 3 + 1 1 3 +.... Use a spreadsheet or a graphing calculator to approimate this sum. Eplain the steps ou used. How man decimal places did ou use in our approimation? Approimating the Natural Base e Work with a partner. Another wa to approimate the natural base e is to consider the epression ( 1 + 1 ). As increases, the value of this epression approaches the value of e. Cop and complete the table. Then use the results in the table to approimate e. Compare this approimation to the one ou obtained in Eploration 1. 10 1 10 10 3 10 10 5 10 ( 1 + 1 ) Graphing a Natural Base Function Work with a partner. Use our approimate value of e in Eploration 1 or to complete the table. Then sketch the graph of the natural base eponential function = e. You can use a graphing calculator and the e ke to check our graph. What are the domain and range of = e? Justif our answers. 1 0 1 = e Communicate Your Answer. What is the natural base e? 5. Repeat Eploration 3 for the natural base eponential function = e. Then compare the graph of = e to the graph of = e.. The natural base e is used in a wide variet of real-life applications. Use the Internet or some other reference to research some of the real-life applications of e. Section 7. The Natural Base e 355
7. Lesson What You Will Learn Core Vocabular natural base e, p. 35 Previous irrational number properties of eponents percent increase percent decrease compound interest Define and use the natural base e. Graph natural base functions. Solve real-life problems. The Natural Base e The histor of mathematics is marked b the discover of special numbers, such as π and i. Another special number is denoted b the letter e. The number is called the natural base e, or the Euler number, after its discoverer, Leonhard Euler (1707 1783). The epression ( 1 + 1 ) approaches e as increases, as shown in the graph and table. 3 1 = e 1 = (1 + 0 0 8 1 ( 10 1 10 10 3 10 10 5 10 ( 1 + 1 ).5937.7081.719.71815.7187.7188 Core Concept The Natural Base e The natural base e is irrational. It is defined as follows: As approaches +, ( 1 + 1 ) approaches e.7188188. Simplifing Natural Base Epressions Check You can use a calculator to check the equivalence of numerical epressions involving e. e^(3)*e^() 8103.08398 e^(9) 8103.08398 Simplif each epression. a. e 3 e b. 1e5 e c. (3e ) a. e 3 e = e 3 + b. 1e5 e = e5 c. (3e ) = 3 (e ) = e 9 = e = 9e 8 = 9 e 8 Monitoring Progress Simplif the epression. Help in English and Spanish at BigIdeasMath.com 1. e 7 e. e8 8e 5 3. (10e 3 ) 3 35 Chapter 7 Eponential and Logarithmic Functions
Graphing Natural Base Functions Core Concept Natural Base Functions A function of the form = ae r is called a natural base eponential function. When a > 0 and r > 0, the function is an eponential growth function. When a > 0 and r < 0, the function is an eponential deca function. The graphs of the basic functions = e and = e are shown. eponential growth 7 5 3 (0, 1) = e (1,.718) 7 5 = e 3 (0, 1) eponential deca (1, 0.38) Graphing Natural Base Functions Tell whether each function represents eponential growth or eponential deca. Then graph the function. ANALYZING MATHEMATICAL RELATIONSHIPS You can rewrite natural base eponential functions to find percent rates of change. In Eample (b), f () = e 0.5 = (e 0.5 ) (0.05) = (1 0.3935). So, the percent decrease is about 39.35%. a. = 3e b. f () = e 0.5 a. Because a = 3 is positive and b. Because a = 1 is positive and r = 1 is positive, the function is r = 0.5 is negative, the function an eponential growth function. is an eponential deca function. Use a table to graph the function. Use a table to graph the function. 1 0 1 0 0.1 1.10 3 8.15 7.39.7 1 0.37 1 (, 7.39) 1 (1, 8.15) 8 ( 1, 1.10) (,.7) (, 0.1) (0, 3) (, 0.37) (0, 1) Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the function represents eponential growth or eponential deca. Then graph the function.. = 1 e 5. = e. f () = e Section 7. The Natural Base e 357
Solving Real-Life Problems You have learned that the balance of an account earning compound interest is given b A = P ( 1 + n) r nt. As the frequenc n of compounding approaches positive infinit, the compound interest formula approimates the following formula. Core Concept Continuousl Compounded Interest When interest is compounded continuousl, the amount A in an account after t ears is given b the formula A = Pe rt where P is the principal and r is the annual interest rate epressed as a decimal. Modeling with Mathematics Balance (dollars) Your Friend s Account A 1,000 10,000 8,000,000,000,000 0 0 (0, 000) 8 1 1 t Year You and our friend each have accounts that earn annual interest compounded continuousl. The balance A (in dollars) of our account after t ears can be modeled b A = 500e 0.0t. The graph shows the balance of our friend s account over time. Which account has a greater principal? Which has a greater balance after 10 ears? 1. Understand the Problem You are given a graph and an equation that represent account balances. You are asked to identif the account with the greater principal and the account with the greater balance after 10 ears.. Make a Plan Use the equation to find our principal and account balance after 10 ears. Then compare these values to the graph of our friend s account. 3. Solve the Problem The equation A = 500e 0.0t is of the form A = Pe rt, where P = 500. So, our principal is $500. Your balance A when t = 10 is A = 500e 0.0(10) = $713.1. Because the graph passes through (0, 000), our friend s principal is $000. The graph also shows that the balance is about $750 when t = 10. ANALYZING MATHEMATICAL RELATIONSHIPS You can also use this reasoning to conclude that our friend s account has a greater annual interest rate than our account. So, our account has a greater principal, but our friend s account has a greater balance after 10 ears.. Look Back Because our friend s account has a lesser principal but a greater balance after 10 ears, the average rate of change from t = 0 to t = 10 should be greater for our friend s account than for our account. Your account: Your friend s account: Monitoring Progress A(10) A(0) 713.1 500 = = 1.31 10 0 10 A(10) A(0) 750 000 = 35 10 0 10 Help in English and Spanish at BigIdeasMath.com 7. You deposit $50 in an account that earns 5% annual interest compounded continuousl. Compare the balance after 10 ears with the accounts in Eample 3. 358 Chapter 7 Eponential and Logarithmic Functions
7. Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY What is the Euler number?. WRITING Tell whether the function f () = 1 3 e represents eponential growth or eponential deca. Eplain. Monitoring Progress and Modeling with Mathematics In Eercises 3 1, simplif the epression. (See Eample 1.) 3. e 3 e5. e e 5. 11e 9 e 10. 7e 7 3e 7. (5e 7 ) 8. (e ) 3 9. 9e 10. 3 8e 1 11. e e e8 1. e e e + 3 ERROR ANALYSIS In Eercises 13 and 1, describe and correct the error in simplifing the epression. 13. 1. (e 3 ) = e (3)() e 5 = e = e 5 e = e 3 In Eercises 15, tell whether the function represents eponential growth or eponential deca. Then graph the function. (See Eample.) 15. = e 3 1. = e 17. = e 18. = 3e 19. = 0.5e 0. = 0.5e 3 1. = 0.e 0.5. = 0.e 0.5 ANALYZING EQUATIONS In Eercises 3, match the function with its graph. Eplain our reasoning. 3. = e. = e 5. = e 0.5. = 0.75e A. B. 8 ( 1, 7.39) C. D. 8 (0, 0.75) 1 (0, 1) (1,.0) ( 1,.59) USING STRUCTURE In Eercises 7 30, use the properties of eponents to rewrite the function in the form = a(1 + r) t or = a(1 r) t. Then find the percent rate of change. 7. = e 0.5t 8. = e 0.75t 9. = e 0.t 30. = 0.5e 0.8t USING TOOLS In Eercises 31 3, use a table of values or a graphing calculator to graph the function. Then identif the domain and range. 8 8 (0, ) (0, 1) (1, 7.39) 31. = e 3. = e + 1 33. = e + 1 3. = 3e 5 Section 7. The Natural Base e 359
35. MODELING WITH MATHEMATICS Investment accounts for a house and education earn annual interest compounded continuousl. The balance H (in dollars) of the house fund after t ears can be modeled b H = 3e 0.05t. The graph shows the balance in the education fund over time. Which account has the greater principal? Which account has a greater balance after 10 ears? (See Eample 3.) H 10,000 Education Account 38. THOUGHT PROVOKING Eplain wh A = P ( 1 + r n ) nt approimates A = Pe rt as n approaches positive infinit. 39. WRITING Can the natural base e be written as a ratio of two integers? Eplain. 0. MAKING AN ARGUMENT Your friend evaluates f () = e when = 1000 and concludes that the graph of = f () has an -intercept at (1000, 0). Is our friend correct? Eplain our reasoning. Balance (dollars) 8,000,000,000,000 (0, 85) 1. DRAWING CONCLUSIONS You invest $500 in an account to save for college. Account 1 pas % annual interest compounded quarterl. Account pas % annual interest compounded continuousl. Which account should ou choose to obtain the greater amount in 10 ears? Justif our answer. 0 0 8 1 1 t Year 3. MODELING WITH MATHEMATICS Tritium and sodium- deca over time. In a sample of tritium, the amount (in milligrams) remaining after t ears is given b = 10e 0.05t. The graph shows the amount of sodium- in a sample over time. Which sample started with a greater amount? Which has a greater amount after 10 ears? Sodium- Deca Amount (milligrams) 0 10 0 0 10 0 t Year 37. OPEN-ENDED Find values of a, b, r, and q such that f () = ae r and g() = be q are eponential deca functions, but f () g() represents eponential growth.. HOW DO YOU SEE IT? Use the graph to complete each statement. a. f () approaches as approaches +. b. f () approaches as approaches. f 3. PROBLEM SOLVING The growth of Mcobacterium tuberculosis bacteria can be modeled b the function N(t) = ae 0.1t, where N is the number of cells after t hours and a is the number of cells when t = 0. a. At 1:00 p.m., there are 30 M. tuberculosis bacteria in a sample. Write a function that gives the number of bacteria after 1:00 p.m. b. Use a graphing calculator to graph the function in part (a). c. Describe how to find the number of cells in the sample at 3:5 p.m. Maintaining Mathematical Proficienc Write the number in scientific notation. (Skills Review Handbook). 0.00 5. 5000.,000,000 7. 0.00000007 Find the inverse of the function. Then graph the function and its inverse. (Section.) 8. = 3 + 5 9. = 1, 0 50. = + 51. = 3 Reviewing what ou learned in previous grades and lessons 30 Chapter 7 Eponential and Logarithmic Functions
7.3 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..B A..C A.5.C Logarithms and Logarithmic Functions Essential Question What are some of the characteristics of the graph of a logarithmic function? Ever eponential function of the form f () = b, where b is a positive real number other than 1, has an inverse function that ou can denote b g() = log b. This inverse function is called a logarithmic function with base b. Rewriting Eponential Equations Work with a partner. Find the value of in each eponential equation. Eplain our reasoning. Then use the value of to rewrite the eponential equation in its equivalent logarithmic form, = log b. a. = 8 b. 3 = 9 c. = d. 5 = 1 e. 5 = 1 5 f. 8 = Graphing Eponential and Logarithmic Functions Work with a partner. Complete each table for the given eponential function. Use the results to complete the table for the given logarithmic function. Eplain our reasoning. Then sketch the graphs of f and g in the same coordinate plane. a. 1 0 1 f () = 10 g () = log 10 1 0 1 b. 1 0 1 f () = e MAKING MATHEMATICAL ARGUMENTS To be proficient in math, ou need to justif our conclusions and communicate them to others. g () = log e 1 0 1 Characteristics of Graphs of Logarithmic Functions Work with a partner. Use the graphs ou sketched in Eploration to determine the domain, range, -intercept, and asmptote of the graph of g() = log b, where b is a positive real number other than 1. Eplain our reasoning. Communicate Your Answer. What are some of the characteristics of the graph of a logarithmic function? 5. How can ou use the graph of an eponential function to obtain the graph of a logarithmic function? Section 7.3 Logarithms and Logarithmic Functions 31
7.3 Lesson What You Will Learn Core Vocabular logarithm of with base b function, p. 3 common logarithm, p. 33 natural logarithm, p. 33 Previous inverse functions Define and evaluate logarithms. Use inverse properties of logarithmic and eponential functions. Graph logarithmic functions. Logarithms You know that = and 3 = 8. However, for what value of does =? Mathematicians define this -value using a logarithm and write = log. The definition of a logarithm can be generalized as follows. Core Concept Definition of Logarithm with Base b Let b and be positive real numbers with b 1. The logarithm of with base b is denoted b log b and is defined as log b = if and onl if b =. The epression log b is read as log base b of. This definition tells ou that the equations log b = and b = are equivalent. The first is in logarithmic form, and the second is in eponential form. Rewriting Logarithmic Equations Rewrite each equation in eponential form. a. log 1 = b. log e 1 = 0 c. log 1 1 = 1 d. log 1/ = 1 Logarithmic Form Eponential Form a. log 1 = = 1 b. log e 1 = 0 e 0 = 1 c. log 1 1 = 1 1 1 = 1 d. log 1/ = 1 ( 1 ) 1 = Rewriting Eponential Equations Rewrite each equation in logarithmic form. a. 5 = 5 b. 10 1 = 0.1 c. 8 /3 = d. 3 = 1 1 Eponential Form Logarithmic Form a. 5 = 5 log 5 5 = b. 10 1 = 0.1 log 10 0.1 = 1 c. 8 /3 = log 8 = 3 d. 3 = 1 1 log 1 1 = 3 3 Chapter 7 Eponential and Logarithmic Functions
Parts (b) and (c) of Eample 1 illustrate two special logarithm values that ou should learn to recognize. Let b be a positive real number such that b 1. Logarithm of 1 Logarithm of b with Base b log b 1 = 0 because b 0 = 1. log b b = 1 because b 1 = b. Evaluating Logarithmic Epressions Evaluate each logarithm. a. log b. log 5 0. c. log 1/5 15 d. log 3 To help ou find the value of log b, ask ourself what power of b gives ou. a. What power of gives ou? 3 =, so log = 3. b. What power of 5 gives ou 0.? 5 1 = 0., so log 5 0. = 1. c. What power of 1 5 gives ou 15? ( 1 5 ) 3 = 15, so log 1/5 15 = 3. d. What power of 3 gives ou? 3 1/ =, so log 3 = 1. A common logarithm is a logarithm with base 10. It is denoted b log 10 or simpl b log. A natural logarithm is a logarithm with base e. It can be denoted b log e but is usuall denoted b ln. Common Logarithm log 10 = log Natural Logarithm log e = ln Evaluating Common and Natural Logarithms Evaluate (a) log 8 and (b) ln 0.3 using a calculator. Round our answer to three decimal places. Check 10^(0.903) 7.998355 e^(-1.0).9999181 Most calculators have kes for evaluating common and natural logarithms. a. log 8 0.903 b. ln 0.3 1.0 Check our answers b rewriting each logarithm in eponential form and evaluating. log(8).903089987 ln(0.3) -1.039780 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Rewrite the equation in eponential form. 1. log 3 81 =. log 7 7 = 1 3. log 1 1 = 0. log 1/ 3 = 5 Rewrite the equation in logarithmic form. 5. 7 = 9. 50 0 = 1 7. 1 = 1 8. 51/8 = Evaluate the logarithm. If necessar, use a calculator and round our answer to three decimal places. 9. log 3 10. log 7 3 11. log 1 1. ln 0.75 Section 7.3 Logarithms and Logarithmic Functions 33
REMEMBER The function g is denoted b f 1, and read as f inverse. Using Inverse Properties B the definition of a logarithm, it follows that the logarithmic function g() = log b is the inverse of the eponential function f () = b. This means that g( f ()) = log b b = and f (g()) = b log b =. In other words, eponential functions and logarithmic functions undo each other. Simplif (a) 10 log and (b) log 5 5. Using Inverse Properties a. 10 log = b logb = b. log 5 5 = log 5 (5 ) Epress 5 as a power with base 5. = log 5 5 Power of a Power Propert = log b b = Finding Inverse Functions Find the inverse of each function. a. f () = b. f () = ln( + 3) a. From the definition of logarithm, the inverse of f () = is f 1 () = log. b. = ln( + 3) Set equal to f(). = ln( + 3) Switch and. e = + 3 Write in eponential form. e 3 = Subtract 3 from each side. The inverse of f () = ln( + 3) is f 1 () = e 3. Check a. f( f 1 ()) = log = b. f 1 ( f ()) = log = f() = ln( + 3) f 1 () = e 3 The graphs appear to be reflections of each other in the line =. Monitoring Progress Simplif the epression. Help in English and Spanish at BigIdeasMath.com 3 Chapter 7 Eponential and Logarithmic Functions 13. 8 log 8 1. log 7 7 3 15. log 1. eln 0 17. Find the inverse of f () =. 18. Find the inverse of f () = ln( 5).
Graphing Logarithmic Functions You can use the inverse relationship between eponential and logarithmic functions to graph logarithmic functions. Core Concept Parent Graphs for Logarithmic Functions The graph of f () = log b is shown below for b > 1 and for 0 < b < 1. Because f () = log b and g() = b are inverse functions, the graph of f () = log b is the reflection of the graph of g() = b in the line =. Graph of f () = log b for b > 1 Graph of f () = log b for 0 < b < 1 g() = b (0, 1) (1, 0) g() = b (0, 1) (1, 0) f() = log b f() = log b Note that the -ais is a vertical asmptote of the graph of f () = log b. Because the range of g() = b is > 0, the domain of its inverse, f () = log b is restricted to > 0. Because the domain of g() = b is all real numbers, the range of its inverse, f() = log b is all real numbers. Graphing a Logarithmic Function Graph f () = log. Identif the domain and range of the function. Step 1 Find the inverse of f. From the definition of logarithm, the inverse of f () = log is f 1 () =. f 1 () 1 1 1 0 1 1 Step Make a table of values for f 1 () =. Step 3 Plot the points from the table and connect them with a smooth curve. Step Because f () = log and f 1 () = are inverse functions, the graph of f is obtained b reflecting the graph of f 1 in the line =. To do this, reverse the coordinates of the points on f 1 and plot these new points on the graph of f. 8 f 1 () = 8 f() = log The domain of f is { > 0} and the range is all real numbers. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Identif the domain and range of the function. 19. f () = log 3 0. f () = log 5 1. f () = log 1/ Section 7.3 Logarithms and Logarithmic Functions 35
7.3 Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE A logarithm with base 10 is called a(n) logarithm.. COMPLETE THE SENTENCE The epression log 3 9 = is read as. 3. WRITING Describe the relationship between f() = 7 and g() = log 7.. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. What power of gives ou 1? What is log base of 1? Evaluate. Evaluate log 1. Monitoring Progress and Modeling with Mathematics In Eercises 5 10, rewrite the equation in eponential form. (See Eample 1.) 5. log 3 9 =. log = 1 7. log 1 = 0 8. log 7 33 = 3 9. log 1/ 1 = 10. log 3 1 3 = 1 In Eercises 11 1, rewrite the equation in logarithmic form. (See Eample.) 11. = 3 1. 1 0 = 1 13. 1 1 = 1 1 1. 5 = 1 5 15. 15 /3 = 5 1. 9 1/ = 7 In Eercises 17, evaluate the logarithm. (See Eample 3.) 17. log 3 81 18. log 7 9 19. log 3 3 0. log 1/ 1 1. log 1 5 5. log 1 8 51 3. log 0.5. log 10 0.001 5. NUMBER SENSE Order the logarithms from least value to greatest value. log 5 3 log 38 log 7 8 log 10. WRITING Eplain wh the epressions log ( 1) and log 1 1 are not defined. In Eercises 7 3, evaluate the logarithm using a calculator. Round our answer to three decimal places. (See Eample.) 7. log 8. ln 1 9. ln 1 3 30. log 7 31. 3 ln 0.5 3. log 0. + 1 33. MODELING WITH MATHEMATICS Skdivers use an instrument called an altimeter to track their altitude as the fall. The altimeter determines altitude b measuring air pressure. The altitude h (in meters) above sea level is related to the air pressure P (in pascals) b the function shown in the diagram. What is the altitude above sea level when the air pressure is 57,000 pascals? P h = 8005 ln 101,300 h = 355 m P = 5,000 Pa h =? P = 57,000 Pa h = 738 m P = 0,000 Pa Not drawn to scale 3. MODELING WITH MATHEMATICS The ph value for a substance measures how acidic or alkaline the substance is. It is given b the formula ph = log[h + ], where H + is the hdrogen ion concentration (in moles per liter). Find the ph of each substance. a. baking soda: [H + ] = 10 8 moles per liter b. vinegar: [H + ] = 10 3 moles per liter 3 Chapter 7 Eponential and Logarithmic Functions
In Eercises 35 0, simplif the epression. (See Eample 5.) 35. 7 log 7 3. 3 log 3 5 37. e ln 38. 10log 15 39. log 3 3 0. ln e + 1 1. ERROR ANALYSIS Describe and correct the error in rewriting 3 = 1 in logarithmic form. log ( 3) = 1. ERROR ANALYSIS Describe and correct the error in simplifing the epression log. log = log (1 ) = log ( ) = log + = + In Eercises 3 5, find the inverse of the function. (See Eample.) 3. f() = 0.3. f() = 11 5. f() = log. f() = log 1/5 7. f() = ln( 1) 8. f() = ln 9. f() = e 3 50. f() = e 51. f() = 5 9 5. f() = 13 + log 53. PROBLEM SOLVING The wind speed s (in miles per hour) near the center of a tornado can be modeled b s = 93 log d + 5, where d is the distance (in miles) that the tornado travels. a. In 195, a tornado traveled 0 miles through three states. Estimate the wind speed near the center of the tornado. b. Find the inverse of the given function. Describe what the inverse represents. 5. MODELING WITH MATHEMATICS The energ magnitude M of an earthquake can be modeled b M = log E 9.9, where E is the amount of energ 3 released (in ergs). Pacific tectonic plate fault line Japan s island Honshu Eurasian tectonic plate a. In 011, a powerful earthquake in Japan, caused b the slippage of two tectonic plates along a fault, released. 10 8 ergs. What was the energ magnitude of the earthquake? b. Find the inverse of the given function. Describe what the inverse represents. In Eercises 55, graph the function. Identif the domain and range of the function. (See Eample 7.) 55. f() = log 5. f() = log 57. f() = log 1/3 58. f() = log 1/ 59. f() = log 0. f() = ln 1. f() = log 1. f() = log 3 ( + ) USING TOOLS In Eercises 3, use a graphing calculator to graph the function. Determine the domain, range, and asmptote of the function. 3. = log( + ). = ln 5. = ln( ). = 3 log 7. MAKING AN ARGUMENT Your friend states that ever logarithmic function will pass through the point (1, 0). Is our friend correct? Eplain our reasoning. 8. ANALYZING RELATIONSHIPS Use the graph of f to determine the domain and range of f 1. Eplain our reasoning. a. f b. f Section 7.3 Logarithms and Logarithmic Functions 37
9. PROBLEM SOLVING Biologists have found that the length (in inches) of an alligator and its weight w (in pounds) are related b the function = 7.1 ln w 3.8. 71. PROBLEM SOLVING A stud in Florida found that the number s of fish species in a pool or lake can be modeled b the function s = 30. 0.5 log A + 3.8(log A) where A is the area (in square meters) of the pool or lake. a. Use a graphing calculator to graph the function. b. Use our graph to estimate the weight of an alligator that is 10 feet long. c. Use the zero feature to find the -intercept of the function. Does this -value make sense in the contet of the situation? Eplain. 70. HOW DO YOU SEE IT? The figure shows the graphs of the two functions f and g. f a. Compare the end behavior of the logarithmic function g to that of the eponential function f. b. Determine whether the functions are inverse functions. Eplain. c. What is the base of each function? Eplain. g a. Use a graphing calculator to graph the function on the domain 00 A 35,000. b. Use our graph to estimate the number of species in a lake with an area of 30,000 square meters. c. Use our graph to estimate the area of a lake that contains si species of fish. d. Describe what happens to the number of fish species as the area of a pool or lake increases. Eplain wh our answer makes sense. 7. THOUGHT PROVOKING Write a logarithmic function that has an output of. Then sketch the graph of our function. 73. CRITICAL THINKING Evaluate each logarithm. (Hint: For each logarithm log b, rewrite b and as powers of the same base.) a. log 15 5 b. log 8 3 c. log 7 81 d. log 18 Maintaining Mathematical Proficienc Let f () = 3. Write a rule for g that represents the indicated transformation of the graph of f. (Section.3) 7. g() = f () 75. g() = f ( 1 ) 7. g() = f ( ) + 3 77. g() = f ( + ) Reviewing what ou learned in previous grades and lessons Identif the function famil to which f belongs. Compare the graph of f to the graph of its parent function. (Section 1.) 78. 79. f 80. 1 f f 38 Chapter 7 Eponential and Logarithmic Functions
7. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.5.A Transformations of Eponential and Logarithmic Functions Essential Question How can ou transform the graphs of eponential and logarithmic functions? Identifing Transformations Work with a partner. Each graph shown is a transformation of the parent function f () = e or f () = ln. Match each function with its graph. Eplain our reasoning. Then describe the transformation of f represented b g. a. g() = e + 3 b. g() = e + + 1 c. g() = e 1 d. g() = ln( + ) e. g() = + ln f. g() = + ln( ) A. B. C. D. E. F. Characteristics of Graphs REASONING To be proficient in math, ou need to make sense of quantities and their relationships in problem situations. Work with a partner. Determine the domain, range, and asmptote of each function in Eploration 1. Justif our answers. Communicate Your Answer 3. How can ou transform the graphs of eponential and logarithmic functions?. Find the inverse of each function in Eploration 1. Then check our answer b using a graphing calculator to graph each function and its inverse in the same viewing window. Section 7. Transformations of Eponential and Logarithmic Functions 39
7. Lesson What You Will Learn Core Vocabular Previous eponential function logarithmic function transformations Transform graphs of eponential functions. Transform graphs of logarithmic functions. Write transformations of graphs of eponential and logarithmic functions. Transforming Graphs of Eponential Functions You can transform graphs of eponential and logarithmic functions in the same wa ou transformed graphs of functions in previous chapters. Eamples of transformations of the graph of f () = are shown below. Core Concept Transformation f() Notation Eamples Horizontal Translation Graph shifts left or right. Vertical Translation Graph shifts up or down. Reflection Graph flips over - or -ais. Horizontal Stretch or Shrink Graph stretches awa from or shrinks toward -ais. Vertical Stretch or Shrink Graph stretches awa from or shrinks toward -ais. f( h) f() + k f( ) f() f(a) a f() g() = 3 g() = + g() = + 5 g() = 1 g() = g() = 3 units right units left 5 units up 1 unit down over -ais over -ais g() = shrink b 1 g() = / stretch b g() = 3( ) stretch b 3 g() = 1 ( ) shrink b 1 Describe the transformation of f () = ( 1 Then graph each function. Translating an Eponential Function ) represented b g() = ( ) 1. STUDY TIP Notice in the graph that the vertical translation also shifted the asmptote units down, so the range of g is { > }. Notice that the function is of the form g() = ( 1 Rewrite the function to identif k. g() = ( 1 ) + ( ) Because k =, the graph of g is a translation units down of the graph of f. k ) g 3 + k. f 3 1 1 3 370 Chapter 7 Eponential and Logarithmic Functions
Translating a Natural Base Eponential Function Describe the transformation of f () = e represented b g() = e + 3 +. Then graph each function. STUDY TIP Notice in the graph that the vertical translation also shifted the asmptote units up, so the range of g is { > }. Notice that the function is of the form g() = e h + k. Rewrite the function to identif h and k. g() = e ( 3) + h k Because h = 3 and k =, the graph of g is a translation 3 units left and units up of the graph of f. g 7 5 3 f ANALYZING MATHEMATICAL RELATIONSHIPS In Eample 3(a), the horizontal shrink follows the translation. In the function h() = 3 3( 5), the translation 5 units right follows the horizontal shrink b a factor of 1 3. Transforming Eponential Functions Describe the transformation of f represented b g. Then graph each function. a. f () = 3, g() = 3 3 5 b. f () = e, g() = 1 8 e a. Notice that the function is of the form g() = 3 a h, where a = 3 and h = 5. So, the graph of g is a translation 5 units right, followed b a horizontal shrink b a factor of 1 3 of the graph of f. b. Notice that the function is of the form g() = ae, where a = 1 8. So, the graph of g is a reflection in the -ais and a vertical shrink b a factor of 1 of the 8 graph of f. 8 f g f g Monitoring Progress Help in English and Spanish at BigIdeasMath.com Describe the transformation of f represented b g. Then graph each function. 1. f () = 10, g() = 10 3 + 1. f () = e, g() = e 5 3. f () = 0., g() = 0.. f () = 10, g() = 10 3 Section 7. Transformations of Eponential and Logarithmic Functions 371
Transforming Graphs of Logarithmic Functions Eamples of transformations of the graph of f () = log are shown below. Core Concept Transformation f() Notation Eamples Horizontal Translation Graph shifts left or right. Vertical Translation Graph shifts up or down. Reflection Graph flips over - or -ais. Horizontal Stretch or Shrink Graph stretches awa from or shrinks toward -ais. Vertical Stretch or Shrink Graph stretches awa from or shrinks toward -ais. f( h) f() + k f( ) f() f(a) a f() g() = log( ) g() = log( + 7) g() = log + 3 g() = log 1 g() = log( ) g() = log units right 7 units left 3 units up 1 unit down over -ais over -ais g() = log() shrink b 1 g() = log ( 1 3 ) stretch b 3 g() = 5 log stretch b 5 g() = 3 log shrink b 3 Transforming Logarithmic Functions Describe the transformation of f represented b g. Then graph each function. a. f () = log, g() = log ( 1 ) b. f () = log 1/, g() = log 1/ ( + ) a. Notice that the function is of the form g() = log(a), where a = 1. STUDY TIP In Eample (b), notice in the graph that the horizontal translation also shifted the asmptote units left, so the domain of g is { > }. So, the graph of g is a reflection in the -ais and a horizontal stretch b a factor of of the graph of f. b. Notice that the function is of the form g() = a log 1/ ( h), where a = and h =. So, the graph of g is a horizontal translation units left and a vertical stretch b a factor of of the graph of f. g 1 8 1 1 8 f 1 1 f g 37 Chapter 7 Eponential and Logarithmic Functions
Monitoring Progress Help in English and Spanish at BigIdeasMath.com Describe the transformation of f represented b g. Then graph each function. 5. f () = log, g() = 3 log. f () = log 1/, g() = log 1/ () 5 Writing Transformations of Graphs of Functions Writing a Transformed Eponential Function Let the graph of g be a reflection in the -ais followed b a translation units right of the graph of f () =. Write a rule for g. Check f Step 1 First write a function h that represents the reflection of f. h() = f () Multipl the output b 1. = Substitute for f (). 5 7 Step Then write a function g that represents the translation of h. h g g() = h ( ) Subtract from the input. = Replace with in h (). The transformed function is g() =. Writing a Transformed Logarithmic Function Let the graph of g be a translation units up followed b a vertical stretch b a factor of of the graph of f () = log 1/3. Write a rule for g. Check 7 g Step 1 First write a function h that represents the translation of f. h() = f () + Add to the output. = log 1/3 + Substitute log 1/3 for f (). 1 h f 1 Step Then write a function g that represents the vertical stretch of h. g() = h() Multipl the output b. = (log 1/3 + ) Substitute log 1/3 + for h(). 3 = log 1/3 + Distributive Propert The transformed function is g() = log 1/3 +. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 7. Let the graph of g be a horizontal stretch b a factor of 3, followed b a translation units up of the graph of f () = e. Write a rule for g. 8. Let the graph of g be a reflection in the -ais, followed b a translation units to the left of the graph of f () = log. Write a rule for g. Section 7. Transformations of Eponential and Logarithmic Functions 373
7. Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Given the function f () = ab h + k, describe the effects of a, h, and k on the graph of the function.. COMPLETE THE SENTENCE The graph of g () = log is a reflection in the of the graph of f () = log. Monitoring Progress and Modeling with Mathematics In Eercises 3, match the function with its graph. Eplain our reasoning. 3. f () = +. g () = + + 5. h () =. k() = + A. B. 1. f () = e, g() = e 15. f () = ( ) 1, g () = ( 1 3 + 1 ) 1. f () = ( 3) 1, g() = ( 1 + 3) 3 In Eercises 17, describe the transformation of f represented b g. Then graph each function. (See Eample 3.) 17. f () = 10, g() = (10) C. D. 1 In Eercises 7 1, describe the transformation of f represented b g. Then graph each function. (See Eamples 1 and.) 7. f () = 3, g() = 3 + 5 8. f () =, g() = 8 9. f () = e, g() = e 1 10. f () = e, g() = e + 11. f () =, g() = 7 1. f () = 10, g() = 10 + 1 3 1 18. f () = e, g () = 3 e 19. f () =, g() = 3() 3 0. f () =, g() = 0.5 5 1. f () = e, g() = 3e. f () = e, g() = e 5 + 3. f () = ( ) 1, g() = ( 1 ) + 5. f () = ( ) 3, g() = ( ) 3 7 + 1 ERROR ANALYSIS In Eercises 5 and, describe and correct the error in graphing the function. 5. f () = + 3 8 13. f () = e, g() = e 37 Chapter 7 Eponential and Logarithmic Functions
. f () = 3 In Eercises 35 38, write a rule for g that represents the indicated transformations of the graph of f. (See Eample 5.) 35. f () = ; translation units down, followed b a reflection in the -ais In Eercises 7 30, describe the transformation of f represented b g. Then graph each function. (See Eample.) 7. f () = log, g() = 3 log + 5 8. f () = log 1/3, g() = log 1/3 ( ) 9. f () = log 1/5, g() = log 1/5 ( 7) 30. f () = log, g() = log ( + ) 3 ANALYZING RELATIONSHIPS In Eercises 31 3, match the function with the correct transformation of the graph of f. Eplain our reasoning. f 3. f () = ( 3 ) ; reflection in the -ais, followed b a vertical stretch b a factor of and a translation units left 37. f () = e ; horizontal shrink b a factor of 1, followed b a translation 5 units up 38. f () = e ; translation units right and 1 unit down, followed b a vertical shrink b a factor of 1 3 In Eercises 39, write a rule for g that represents the indicated transformation of the graph of f. (See Eample.) 39. f () = log ; vertical stretch b a factor of, followed b a translation 5 units down 0. f () = log ; reflection in the -ais, followed b a translation 9 units right 1. f () = log ; translation 3 units right and units up, followed b a reflection in the -ais. f () = ln ; translation 3 units right and 1 unit up, followed b a vertical stretch b a factor of 8 31. = f ( ) 3. = f ( + ) 33. = f () 3. = f () A. B. JUSTIFYING STEPS In Eercises 3 and, justif each step in writing a rule for g that represents the indicated transformations of the graph of f. 3. f () = ln ; reflection in the -ais, followed b a translation units down h () = f () = ln g () = h () = ln. f () = 8 ; vertical stretch b a factor of, followed b a translation 1 unit up and 3 units left C. D. h() = f () = 8 g () = h( + 3) + 1 = 8 + 3 + 1 Section 7. Transformations of Eponential and Logarithmic Functions 375
USING STRUCTURE In Eercises 5 8, describe the transformation of the graph of f represented b the graph of g. Then give an equation of the asmptote. 5. f () = e, g() = e +. f () = 10, g() = 10 5 7. f () = ln, g() = ln( + ) 8. f () = log 1/5, g() = log 1/5 + 13 9. MODELING WITH MATHEMATICS The slope S of a beach is related to the average diameter d (in millimeters) of the sand particles on the beach b the equation S = 0.159 + 0.118 log d. Describe the transformation of f (d ) = log d represented b S. Then use the function to determine the slope of a beach for each sand tpe below. Sand particle Diameter (mm), d fine sand 0.15 51. MAKING AN ARGUMENT Your friend claims a single transformation of f () = log can result in a function g whose graph never intersects the graph of f. Is our friend correct? Eplain our reasoning. 5. THOUGHT PROVOKING Is it possible to transform the graph of f () = e to obtain the graph of g() = ln? Eplain our reasoning. 53. ABSTRACT REASONING Determine whether each statement is alwas, sometimes, or never true. Eplain our reasoning. a. A vertical translation of the graph of f () = log changes the equation of the asmptote. b. A vertical translation of the graph of f () = e changes the equation of the asmptote. c. A horizontal shrink of the graph of f () = log does not change the domain. d. The graph of g() = ab h + k does not intersect the -ais. medium sand 0.5 coarse sand 0.5 ver coarse sand 1 5. PROBLEM SOLVING The amount P (in grams) of 100 grams of plutonium-39 that remains after t ears can be modeled b P = 100(0.99997) t. a. Describe the domain and range of the function. 50. HOW DO YOU SEE IT? The graphs of f () = b b) and g() = ( 1 are shown for b =. g 8 f b. How much plutonium-39 is present after 1,000 ears? c. Describe the transformation of the function if the initial amount of plutonium were 550 grams. d. Does the transformation in part (c) affect the domain and range of the function? Eplain our reasoning. a. Use the graph to describe a transformation of the graph of f that results in the graph of g. b. Does our answer in part (a) change when 0 < b < 1? Eplain. Maintaining Mathematical Proficienc Perform the indicated operation. (Section.5) 57. Let f () = and g() =. Find ( fg)(). Then evaluate the product when = 3. g) 58. Let f () = and g() = 3. Find ( f (). Then evaluate the quotient when = 5. 55. CRITICAL THINKING Consider the graph of the function h () = e. Describe the transformation of the graph of f () = e represented b the graph of h. Then describe the transformation of the graph of g() = e represented b the graph of h. Justif our answers. 5. OPEN-ENDED Write a function of the form = ab h + k whose graph has a -intercept of 5 and an asmptote of =. Reviewing what ou learned in previous grades and lessons 59. Let f () = 3 and g() = 8 3. Find ( f + g)(). Then evaluate the sum when =. 0. Let f () = and g() = 3. Find ( f g)(). Then evaluate the difference when =. 37 Chapter 7 Eponential and Logarithmic Functions
7.1 7. What Did You Learn? Core Vocabular eponential function, p. 38 eponential growth function, p. 38 growth factor, p. 38 asmptote, p. 38 eponential deca function, p. 38 deca factor, p. 38 natural base e, p. 35 logarithm of with base b function, p. 3 common logarithm, p. 33 natural logarithm, p. 33 Core Concepts Section 7.1 Parent Function for Eponential Growth Functions, p. 38 Parent Function for Eponential Deca Functions, p. 38 Section 7. The Natural Base e, p. 35 Natural Base Functions, p. 357 Eponential Growth and Deca Models, p. 39 Compound Interest, p. 351 Continuousl Compounded Interest, p. 358 Section 7.3 Definition of Logarithm with Base b, p. 3 Parent Graphs for Logarithmic Functions, p. 35 Section 7. Transforming Graphs of Eponential Functions, p. 370 Transforming Graphs of Logarithmic Functions, p. 37 Mathematical Thinking 1. How did ou check to make sure our answer was reasonable in Eercise 3 on page 35?. How can ou justif our conclusions in Eercises 3 on page 359? 3. How did ou monitor and evaluate our progress in Eercise on page 37? Stud Skills Forming a Weekl Stud Group Select students who are just as dedicated to doing well in the math class as ou are. Find a regular meeting place that has minimal distractions. Compare schedules and plan at least one time a week to meet, allowing at least 1.5 hours for stud time. 377
7.1 7. Quiz Tell whether the function represents eponential growth or eponential deca. Eplain our reasoning. (Section 7.1 and Section 7.) 1. f () = (.5). = ( 3 8) Simplif the epression. (Section 7.1 and Section 7.) 5. e 8 e. 15e 3 3e 3. = e 0.. f () = 5e 7. (5e ) 3 8. e ln 9 9. log 7 9 10. log 3 81 Rewrite the epression in eponential or logarithmic form. (Section 7.3) 11. log 10 = 5 1. log 1/3 7 = 3 13. 7 = 01 1. = 0.05 Evaluate the logarithm. If necessar, use a calculator and round our answer to three decimal places. (Section 7.3) 15. log 5 1. ln 1. 17. log 3 Graph the function and its inverse. Identif the domain and range of the function and its inverse. (Section 7.3) 18. f () = ( 1 9) 19. f() = ln( 7) 0. f () = log 5 ( + 1) The graph of g is a transformation of the graph of f. Write a rule for g. (Section 7.) 1. f () = log 3. f () = 3 3. f () = log 1/ 3 1 8 1 g g g. You purchase an antique lamp for $150. The value of the lamp increases b.15% each ear. Write an eponential model that gives the value (in dollars) of the lamp t ears after ou purchased it. (Section 7.1) 5. A local bank advertises two certificate of deposit (CD) accounts that ou can use to save mone and earn interest. The interest is compounded monthl for both accounts. (Section 7.1) a. You deposit the minimum required amounts in each CD account. How much mone is in each account at the end of its term? How much interest does each account earn? Justif our answers. b. Describe the benefits and drawbacks of each account. CD Specials.0 % APY 3/mo CD $1500 Minimum Balance 3.0 % APY 0/mo CD $000 Minimum Balance. The Richter scale is used for measuring the magnitude of an earthquake. The Richter magnitude R is given b R = 0.7 ln E + 1.17, where E is the energ (in kilowatt-hours) released b the earthquake. Graph the model. What is the Richter magnitude for an earthquake that releases 3,000 kilowatt-hours of energ? (Section 7.) 378 Chapter 7 Eponential and Logarithmic Functions
7.5 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.5.C Preparing for A.5.D MAKING MATHEMATICAL ARGUMENTS To be proficient in math, ou need to understand and use stated assumptions, definitions, and previousl established results. Properties of Logarithms Essential Question How can ou use properties of eponents to derive properties of logarithms? Let = log b m and = log b n. The corresponding eponential forms of these two equations are b = m and b = n. Product Propert of Logarithms Work with a partner. To derive the Product Propert, multipl m and n to obtain mn = b b = b +. The corresponding logarithmic form of mn = b + is log b mn = +. So, log b mn =. Product Propert of Logarithms Quotient Propert of Logarithms Work with a partner. To derive the Quotient Propert, divide m b n to obtain m n = b b = b. The corresponding logarithmic form of m n = b is log m b =. So, n log m b =. Quotient Propert of Logarithms n Power Propert of Logarithms Work with a partner. To derive the Power Propert, substitute b for m in the epression log b m n, as follows. log b m n = log b (b ) n Substitute b for m. = log b b n Power of a Power Propert of Eponents = n Inverse Propert of Logarithms So, substituting log b m for, ou have log b m n =. Power Propert of Logarithms Communicate Your Answer. How can ou use properties of eponents to derive properties of logarithms? 5. Use the properties of logarithms that ou derived in Eplorations 1 3 to evaluate each logarithmic epression. a. log 1 3 b. log 3 81 3 c. ln e + ln e 5 d. ln e ln e 5 e. log 5 75 log 5 3 f. log + log 3 Section 7.5 Properties of Logarithms 379
7.5 Lesson What You Will Learn Core Vocabular Previous base properties of eponents STUDY TIP These three properties of logarithms correspond to these three properties of eponents. a m a n = a m + n a m a n = am n (a m ) n = a mn Use the properties of logarithms to evaluate logarithms. Use the properties of logarithms to epand or condense logarithmic epressions. Use the change-of-base formula to evaluate logarithms. Properties of Logarithms You know that the logarithmic function with base b is the inverse function of the eponential function with base b. Because of this relationship, it makes sense that logarithms have properties similar to properties of eponents. Core Concept Properties of Logarithms Let b, m, and n be positive real numbers with b 1. Product Propert log b mn = log b m + log b n Quotient Propert log b m n = log b m log b n Power Propert log b m n = n log b m Using Properties of Logarithms Use log 3 1.585 and log 7.807 to evaluate each logarithm. a. log 3 7 b. log 1 c. log 9 COMMON ERROR Note that in general log m b n log b m log b n and log b mn (log b m)(log b n). a. log 3 7 = log 3 log 7 Quotient Propert 1.585.807 Use the given values of log 3 and log 7. = 1. Subtract. b. log 1 = log (3 7) Write 1 as 3 7. = log 3 + log 7 Product Propert 1.585 +.807 Use the given values of log 3 and log 7. =.39 Add. c. log 9 = log 7 Write 9 as 7. = log 7 Power Propert (.807) Use the given value log 7. = 5.1 Multipl. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use log 5 0.898 and log 8 1.11 to evaluate the logarithm. 1. log 5 8. log 0 3. log. log 15 380 Chapter 7 Eponential and Logarithmic Functions
Rewriting Logarithmic Epressions You can use the properties of logarithms to epand and condense logarithmic epressions. Epanding a Logarithmic Epression STUDY TIP When ou are epanding or condensing an epression involving logarithms, ou can assume that an variables are positive. Epand ln 57. ln 57 = ln 57 ln Quotient Propert = ln 5 + ln 7 ln Product Propert = ln 5 + 7 ln ln Power Propert Condense log 9 + 3 log log 3. Condensing a Logarithmic Epression log 9 + 3 log log 3 = log 9 + log 3 log 3 Monitoring Progress Power Propert = log(9 3 ) log 3 Product Propert = log 9 3 3 = log Simplif. Epand the logarithmic epression. 5. log 3 5. ln 1 Condense the logarithmic epression. Quotient Propert Help in English and Spanish at BigIdeasMath.com 7. log log 9 8. ln + 3 ln 3 ln 1 Change-of-Base Formula Logarithms with an base other than 10 or e can be written in terms of common or natural logarithms using the change-of-base formula. This allows ou to evaluate an logarithm using a calculator. Core Concept Change-of-Base Formula If a, b, and c are positive real numbers with b 1 and c 1, then log c a = log b a log b c. In particular, log c a = log a log c and log c a = ln a ln c. Section 7.5 Properties of Logarithms 381
Changing a Base Using Common Logarithms ANOTHER WAY In Eample, log 3 8 can be evaluated using natural logarithms. log 3 8 = ln 8 ln 3 1.893 Notice that ou get the same answer whether ou use natural logarithms or common logarithms in the change-of-base formula. Evaluate log 3 8 using common logarithms. log 3 8 = log 8 log 3 log c a = log a log c 0.9031 1.893 Use a calculator. Then divide. 0.771 Evaluate log using natural logarithms. ln log = ln Changing a Base Using Natural Logarithms log c a = ln a ln c 3.1781 1.77 Use a calculator. Then divide. 1.7918 Solving a Real-Life Problem For a sound with intensit I (in watts per square meter), the loudness L(I ) of the sound (in decibels) is given b the function L(I) = 10 log I I 0 where I 0 is the intensit of a barel audible sound (about 10 1 watts per square meter). An artist in a recording studio turns up the volume of a track so that the intensit of the sound doubles. B how man decibels does the loudness increase? Let I be the original intensit, so that I is the doubled intensit. increase in loudness = L(I ) L(I ) Write an epression. = 10 log I 10 log I I 0 I 0 = 10 ( log I log I I 0 I 0 ) = 10 ( log + log I log I I 0 I 0 ) Substitute. = 10 log Simplif. The loudness increases b 10 log decibels, or about 3 decibels. Monitoring Progress Use the change-of-base formula to evaluate the logarithm. Distributive Propert Product Propert Help in English and Spanish at BigIdeasMath.com 9. log 5 8 10. log 8 1 11. log 9 1. log 1 30 13. WHAT IF? In Eample, the artist turns up the volume so that the intensit of the sound triples. B how man decibels does the loudness increase? 38 Chapter 7 Eponential and Logarithmic Functions
7.5 Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE To condense the epression log 3 + log 3, ou need to use the Propert of Logarithms.. WRITING Describe two was to evaluate log 7 1 using a calculator. Monitoring Progress and Modeling with Mathematics In Eercises 3 8, use log 7 0.71 and log 7 1 1.77 to evaluate the logarithm. (See Eample 1.) 3. log 7 3. log 7 8. ln 8 3 = 3 ln 8 + ln 5. log 7 1. log 7 7. log 7 1 8. log 7 1 3 In Eercises 9 1, match the epression with the logarithm that has the same value. Justif our answer. 9. log 3 log 3 A. log 3 10. log 3 B. log 3 3 11. log 3 C. log 3 1 1. log 3 + log 3 D. log 3 3 In Eercises 13 0, epand the logarithmic epression. (See Eample.) 13. log 3 1. log 8 3 15. log 10 5 1. ln 3 17. ln 3 18. ln 19. log 7 5 0. log 5 3 ERROR ANALYSIS In Eercises 1 and, describe and correct the error in epanding the logarithmic epression. 1. log 5 = (log 5)(log ) In Eercises 3 30, condense the logarithmic epression. (See Eample 3.) 3. log 7 log 10. ln 1 ln 5. ln + ln. log + log 11 7. log 5 + 1 3 log 5 8. ln ln 9. 5 ln + 7 ln + ln 30. log 3 + log 3 1 + log 3 31. REASONING Which of the following is not equivalent to log 5? Justif our answer. 3 A B C D log 5 log 5 3 log 5 log 5 3 + log 5 log 5 log 5 3 log 5 log 5 log 5 3 log 5 3. REASONING Which of the following equations is correct? Justif our answer. A log 7 + log 7 = log 7 ( + ) B 9 log log = log 9 C 5 log + 7 log = log 5 7 D log 9 5 log 9 = log 9 5 Section 7.5 Properties of Logarithms 383
In Eercises 33 0, use the change-of-base formula to evaluate the logarithm. (See Eamples and 5.) 33. log 7 3. log 5 13 35. log 9 15 3. log 8. The intensit of the sound of a certain television advertisement is 10 times greater than the intensit of the television program. B how man decibels does the loudness increase? Intensit of Television Sound 37. log 17 38. log 8 39. log 7 3 1 0. log 3 9 0 1. MAKING AN ARGUMENT Your friend claims ou can use the change-of-base formula to graph = log 3 using a graphing calculator. Is our friend correct? Eplain our reasoning.. HOW DO YOU SEE IT? Use the graph to determine the value of log 8 log. = log 8 MODELING WITH MATHEMATICS In Eercises 3 and, use the function L(I ) given in Eample. 3. The blue whale can produce sound with an intensit that is 1 million times greater than the intensit of the loudest sound a human can make. Find the difference in the decibel levels of the sounds made b a blue whale and a human. (See Eample.) During show: Intensit = I During ad: Intensit = 10I 5. REWRITING A FORMULA Under certain conditions, the wind speed s (in knots) at an altitude of h meters above a grass plain can be modeled b the function s(h) = ln 100h. a. B what amount does the wind speed increase when the altitude doubles? b. Show that the given function can be written in terms of common logarithms as s(h) = (log h + ). log e. THOUGHT PROVOKING Determine whether the formula log b (M + N) = log b M + log b N is true for all positive, real values of M, N, and b (with b 1). Justif our answer. 7. USING STRUCTURE Use the properties of eponents to prove the change-of-base formula. (Hint: Let = log b a, = log b c, and z = log c a.) 8. CRITICAL THINKING Describe three was to transform the graph of f () = log to obtain the graph of g() = log 100 1. Justif our answers. Maintaining Mathematical Proficienc Solve the inequalit b graphing. (Section.) Reviewing what ou learned in previous grades and lessons 9. > 0 50. ( ) 5 37 51. + 13 + < 0 5. + Solve the equation b graphing the related sstem of equations. (Section.5) 53. 3 = + 5 + 3 5. ( + 3)( ) = 55. 5 = ( + 3) + 10 5. ( + 7) + 5 = ( + 10) 3 38 Chapter 7 Eponential and Logarithmic Functions
7. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.5.D A.5.E Solving Eponential and Logarithmic Equations Essential Question How can ou solve eponential and logarithmic equations? Solving Eponential and Logarithmic Equations Work with a partner. Match each equation with the graph of its related sstem of equations. Eplain our reasoning. Then use the graph to solve the equation. a. e = b. ln = 1 c. = 3 d. log = 1 e. log 5 = 1 f. = A. B. C. D. E. F. USING PROBLEM-SOLVING STRATEGIES To be proficient in math, ou need to plan a solution pathwa rather than simpl jumping into a solution attempt. Solving Eponential and Logarithmic Equations Work with a partner. Look back at the equations in Eplorations 1(a) and 1(b). Suppose ou want a more accurate wa to solve the equations than using a graphical approach. a. Show how ou could use a numerical approach b creating a table. For instance, ou might use a spreadsheet to solve the equations. b. Show how ou could use an analtical approach. For instance, ou might tr solving the equations b using the inverse properties of eponents and logarithms. Communicate Your Answer 3. How can ou solve eponential and logarithmic equations?. Solve each equation using an method. Eplain our choice of method. a. 1 = b. = + 1 c. = 3 + 1 d. log = 1 e. ln = f. log 3 = 3 Section 7. Solving Eponential and Logarithmic Equations 385
7. Lesson What You Will Learn Core Vocabular eponential equations, p. 38 logarithmic equations, p. 387 Previous etraneous solution inequalit Solve eponential equations. Solve logarithmic equations. Solve eponential and logarithmic inequalities. Solving Eponential Equations Eponential equations are equations in which variable epressions occur as eponents. The result below is useful for solving certain eponential equations. Core Concept Propert of Equalit for Eponential Equations Algebra If b is a positive real number other than 1, then b = b if and onl if =. Eample If 3 = 3 5, then = 5. If = 5, then 3 = 3 5. The preceding propert is useful for solving an eponential equation when each side of the equation uses the same base (or can be rewritten to use the same base). When it is not convenient to write each side of an eponential equation using the same base, ou can tr to solve the equation b taking a logarithm of each side. Solving Eponential Equations Solve each equation. a. 100 = ( 10) 1 3 b. = 7 Check 100 1 =? ( 10) 1 1 3 100 =? ( 10) 1 100 = 100 a. 100 = ( 10) 1 3 Write original equation. (10 ) = (10 1 ) 3 Rewrite 100 and 1 as powers with base 10. 10 10 = 10 + 3 Power of a Power Propert = + 3 Propert of Equalit for Eponential Equations = 1 Solve for. b. = 7 Write original equation. log = log 7 Take log of each side. = log 7 log b b =.807 Use a calculator. Check Enter = and = 7 in a graphing calculator. Use the intersect feature to find the intersection point of the graphs. The graphs intersect at about (.807, 7). So, the solution of = 7 is about.807. 10 0 Intersection X=.807359 Y=7 3 5 38 Chapter 7 Eponential and Logarithmic Functions
An important application of eponential equations is Newton s Law of Cooling. This law states that for a cooling substance with initial temperature T 0, the temperature T after t minutes can be modeled b ANALYZING MATHEMATICAL RELATIONSHIPS Notice that Newton's Law of Cooling models the temperature of a cooling bod b adding a constant function, T R, to a decaing eponential function, (T 0 T R )e rt. T = (T 0 T R )e rt + T R where T R is the surrounding temperature and r is the cooling rate of the substance. Solving a Real-Life Problem You are cooking aleecha, an Ethiopian stew. When ou take it off the stove, its temperature is 1 F. The room temperature is 70 F, and the cooling rate of the stew is r = 0.0. How long will it take to cool the stew to a serving temperature of 100 F? Use Newton s Law of Cooling with T = 100, T 0 = 1, T R = 70, and r = 0.0. T = (T 0 T R )e rt + T R Newton s Law of Cooling 100 = (1 70)e 0.0t + 70 Substitute for T, T 0, T R, and r. 30 = 1e 0.0t Subtract 70 from each side. 0.11 e 0.0t Divide each side b 1. ln 0.11 ln e 0.0t Take natural log of each side. 1.55 0.0t ln e = log e e = 33.8 t Divide each side b 0.0. You should wait about 3 minutes before serving the stew. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation. 1. = 5. 7 9 = 15 3. e 0.3 7 = 13. WHAT IF? In Eample, how long will it take to cool the stew to 100ºF when the room temperature is 75ºF? Solving Logarithmic Equations Logarithmic equations are equations that involve logarithms of variable epressions. You can use the net propert to solve some tpes of logarithmic equations. Core Concept Propert of Equalit for Logarithmic Equations Algebra If b,, and are positive real numbers with b 1, then log b = log b if and onl if =. Eample If log = log 7, then = 7. If = 7, then log = log 7. The preceding propert implies that if ou are given an equation =, then ou can eponentiate each side to obtain an equation of the form b = b. This technique is useful for solving some logarithmic equations. Section 7. Solving Eponential and Logarithmic Equations 387
Solving Logarithmic Equations Solve (a) ln( 7) = ln( + 5) and (b) log (5 17) = 3. Check ln( 7) =? ln( + 5) ln(1 7) =? ln 9 ln 9 = ln 9 a. ln( 7) = ln( + 5) Write original equation. 7 = + 5 Propert of Equalit for Logarithmic Equations 3 7 = 5 Subtract from each side. 3 = 1 Add 7 to each side. = Divide each side b 3. b. log (5 17) = 3 Write original equation. Check log (5 5 17) =? 3 log (5 17) =? 3 log 8 =? 3 Because 3 = 8, log 8 = 3. log (5 17) = 3 Eponentiate each side using base. 5 17 = 8 5 = 5 b log b = Add 17 to each side. = 5 Divide each side b 5. Because the domain of a logarithmic function generall does not include all real numbers, be sure to check for etraneous solutions of logarithmic equations. You can do this algebraicall or graphicall. Solve log + log( 5) =. Solving a Logarithmic Equation Check log( 10) + log(10 5) =? log 0 + log 5 =? log 100 =? = log[ ( 5)] + log( 5 5) =? log( 10) + log( 10) =? Because log( 10) is not defined, 5 is not a solution. log + log( 5) = Write original equation. log[( 5)] = Product Propert of Logarithms 10 log[( 5)] = 10 Eponentiate each side using base 10. ( 5) = 100 b log b = 10 = 100 Distributive Propert 10 100 = 0 Write in standard form. 5 50 = 0 Divide each side b. ( 10)( + 5) = 0 Factor. = 10 or = 5 Zero-Product Propert The apparent solution = 5 is etraneous. So, the onl solution is = 10. Monitoring Progress Solve the equation. Check for etraneous solutions. Help in English and Spanish at BigIdeasMath.com 5. ln(7 ) = ln( + 11). log ( ) = 5 7. log 5 + log( 1) = 8. log ( + 1) + log = 3 388 Chapter 7 Eponential and Logarithmic Functions
STUDY TIP Be sure ou understand that these properties of inequalit are onl true for values of b > 1. Solving Eponential and Logarithmic Inequalities Eponential inequalities are inequalities in which variable epressions occur as eponents, and logarithmic inequalities are inequalities that involve logarithms of variable epressions. To solve eponential and logarithmic inequalities algebraicall, use these properties. Note that the properties are true for and. Eponential Propert of Inequalit: If b is a positive real number greater than 1, then b > b if and onl if >, and b < b if and onl if <. Logarithmic Propert of Inequalit: If b,, and are positive real numbers with b > 1, then log b > log b if and onl if >, and log b < log b if and onl if <. You can also solve an inequalit b taking a logarithm of each side or b eponentiating. Solving an Eponential Inequalit Solve 3 < 0. 3 < 0 Write original inequalit. log 3 3 < log 3 0 < log 3 0 Take log 3 of each side. log b b = The solution is < log 3 0. Because log 3 0.77, the approimate solution is <.77. Solve log. Solving a Logarithmic Inequalit Method 1 Use an algebraic approach. log Write original inequalit. 10 log 10 10 Eponentiate each side using base 10. 100 b log b = Because log is onl defined when > 0, the solution is 0 < 100. Method Use a graphical approach. Graph = log and = in the same viewing window. Use the intersect feature to determine that the graphs intersect when = 100. The graph of = log is on or below the graph of = when 0 < 100. The solution is 0 < 100. Monitoring Progress Solve the inequalit. 50 Intersection X=100 Y= 1 Help in English and Spanish at BigIdeasMath.com 9. e < 10. 10 > 3 11. log + 9 < 5 1. ln 1 > 3 175 Section 7. Solving Eponential and Logarithmic Equations 389
7. Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The equation 3 1 = 3 is an eample of a(n) equation.. WRITING Compare the methods for solving eponential and logarithmic equations. 3. WRITING When do logarithmic equations have etraneous solutions?. COMPLETE THE SENTENCE If b is a positive real number other than 1, then b = b if and onl if. Monitoring Progress and Modeling with Mathematics In Eercises 5 1, solve the equation. (See Eample 1.) 5. 7 3 + 5 = 7 1. e = e 3 1 7. 5 3 = 5 5 8. = 3 3 5 9. 3 = 7 10. 5 = 33 11. 9 5 + = ( 1 11 7) 1. 51 5 1 = ( 8) 1 13. 7 5 = 1 1. 11 = 38 In Eercises 19 and 0, use Newton s Law of Cooling to solve the problem. (See Eample.) 19. You are driving on a hot da when our car overheats and stops running. The car overheats at 80 F and can be driven again at 30 F. When it is 80 F outside, the cooling rate of the car is r = 0.0058. How long do ou have to wait until ou can continue driving? 15. 3e + 9 = 15 1. e 7 = 5 17. MODELING WITH MATHEMATICS The length (in centimeters) of a scalloped hammerhead shark can be modeled b the function = 19e 0.05t where t is the age (in ears) of the shark. How old is a shark that is 175 centimeters long? 0. You cook a turke until the internal temperature reaches 180 F. The turke is placed on the table until the internal temperature reaches 100 F and it can be carved. When the room temperature is 7 F, the cooling rate of the turke is r = 0.07. How long do ou have to wait until ou can carve the turke? In Eercises 1 3, solve the equation. (See Eample 3.) 1. ln( 7) = ln( + 11). ln( ) = ln( + ) 3. log (3 ) = log 5. log(7 + 3) = log 38 18. MODELING WITH MATHEMATICS One hundred grams of radium are stored in a container. The amount R (in grams) of radium present after t ears can be modeled b R = 100e 0.0003t. After how man ears will onl 5 grams of radium be present? 5. log ( + 8) = 5. log 3 ( + 1) = 7. log 7 ( + 9) = 8. log 5 (5 + 10) = 9. log(1 9) = log 3 30. log (5 + 9) = log 31. log ( ) = 3. log 3 ( + 9 + 7) = 390 Chapter 7 Eponential and Logarithmic Functions
In Eercises 33 0, solve the equation. Check for etraneous solutions. (See Eample.) 33. log + log ( ) = 3 3. log 3 + log ( 1) = 3 35. ln + ln( + 3) = 3. ln + ln( ) = 5 5. ANALYZING RELATIONSHIPS Approimate the solution of each equation using the graph. a. 1 5 5 = 9 b. log 5 = 8 = 9 1 8 = 37. log 3 3 + log 3 3 = 38. log ( ) + log ( + 10) = 1 = 1 5 5 = log 5 39. log 3 ( 9) + log 3 ( 3) = 0. log 5 ( + ) + log 5 ( + 1) = ERROR ANALYSIS In Eercises 1 and, describe and correct the error in solving the equation. 1.. log 3 (5 1) = 3 log 3 (5 1) = 3 5 1 = 5 = 5 = 13 log ( + 1) + log = 3 log [( + 1)()] = 3 log [( + 1)()] = 3 ( + 1)() = + 1 = 0 ( + 1)( ) = 0 = 1 or = 3. PROBLEM SOLVING You deposit $100 in an account that pas % annual interest. How long will it take for the balance to reach $1000 for each frequenc of compounding? a. annuall b. quarterl c. dail d. continuousl. MODELING WITH MATHEMATICS The apparent magnitude of a star is a measure of the brightness of the star as it appears to observers on Earth. The apparent magnitude M of the dimmest star that can be seen with a telescope is M = 5 log D +, where D is the diameter (in millimeters) of the telescope s objective lens. What is the diameter of the objective lens of a telescope that can reveal stars with a magnitude of 1?. MAKING AN ARGUMENT Your friend states that a logarithmic equation cannot have a negative solution because logarithmic functions are not defined for negative numbers. Is our friend correct? Justif our answer. In Eercises 7 5, solve the inequalit. (See Eamples 5 and.) 7. 9 > 5 8. 3 9. ln 3 50. log < 51. 3 5 < 8 5. e 3 + > 11 53. 3 log 5 + 9 5. log 5 5 3 55. COMPARING METHODS Solve log 5 < algebraicall and graphicall. Which method do ou prefer? Eplain our reasoning. 5. PROBLEM SOLVING You deposit $1000 in an account that pas 3.5% annual interest compounded monthl. When is our balance at least $100? $3500? 57. PROBLEM SOLVING An investment that earns a rate of return r doubles in value in t ears, where ln t = and r is epressed as a decimal. What ln(1 + r) rates of return will double the value of an investment in less than 10 ears? 58. PROBLEM SOLVING Your famil purchases a new car for $0,000. Its value decreases b 15% each ear. During what interval does the car s value eceed $10,000? USING TOOLS In Eercises 59, use a graphing calculator to solve the equation. 59. ln = 3 + 0. log = 7 1. log = 3 3. ln = e 3 Section 7. Solving Eponential and Logarithmic Equations 391
3. REWRITING A FORMULA A biologist can estimate the age of an African elephant b measuring the length of its footprint and using the equation = 5 5.7e 0.09a, where is the length 3 cm (in centimeters) of the footprint and a is the age (in ears). a. Rewrite the equation, solving for a in terms of. b. Use the equation in part (a) to find the ages of the elephants whose footprints are shown. 3 cm 8 cm cm. HOW DO YOU SEE IT? Use the graph to solve the inequalit ln + > 9. Eplain our reasoning. 1 CRITICAL THINKING In Eercises 7 7, solve the equation. 7. + 3 = 5 3 1 8. 10 3 8 = 5 9. log 3 ( ) = log 9 70. log = log 8 71. 1 + 3 = 0 7. 5 + 0 5 15 = 0 73. WRITING In Eercises 7 70, ou solved eponential and logarithmic equations with different bases. Describe general methods for solving such equations. 7. PROBLEM SOLVING When X-ras of a fied wavelength strike a material centimeters thick, the intensit I() of the X-ras transmitted through the material is given b I() = I 0 e μ, where I 0 is the initial intensit and μ is a value that depends on the tpe of material and the wavelength of the X-ras. The table shows the values of μ for various materials and X-ras of medium wavelength. = 9 = ln + Material Aluminum Copper Lead Value of μ 0.3 3. 3 5. OPEN-ENDED Write an eponential equation that has a solution of =. Then write a logarithmic equation that has a solution of = 3.. THOUGHT PROVOKING Give eamples of logarithmic or eponential equations that have one solution, two solutions, and no solutions. a. Find the thickness of aluminum shielding that reduces the intensit of X-ras to 30% of their initial intensit. (Hint: Find the value of for which I() = 0.3I 0.) b. Repeat part (a) for the copper shielding. c. Repeat part (a) for the lead shielding. d. Your dentist puts a lead apron on ou before taking X-ras of our teeth to protect ou from harmful radiation. Based on our results from parts (a) (c), eplain wh lead is a better material to use than aluminum or copper. Maintaining Mathematical Proficienc Write an equation in point-slope form of the line that passes through the given point and has the given slope. (Skills Review Handbook) 75. (1, ); m = 7. (3, ); m = Reviewing what ou learned in previous grades and lessons 1 3 77. (3, 8); m = 78. (, 5); m = Use finite differences to determine the degree of the polnomial function that fits the data. Then use technolog to find the polnomial function. (Section 5.9) 79. ( 3, 50), (, 13), ( 1, 0), (0, 1), (1, ), (, 15), (3, 5), (, 15) 80. ( 3, 139), (, 3), ( 1, 1), (0, ), (1, 1), (, ), (3, 37), (, 1) 81. ( 3, 37), (, 8), ( 1, 17), (0, ), (1, 3), (, 3), (3, 189), (, ) 39 Chapter 7 Eponential and Logarithmic Functions
7.7 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.5.B A.8.A A.8.B A.8.C Modeling with Eponential and Logarithmic Functions Essential Question How can ou recognize polnomial, eponential, and logarithmic models? Recognizing Different Tpes of Models Work with a partner. Match each tpe of model with the appropriate scatter plot. Use a regression program to find a model that fits the scatter plot. a. linear (positive slope) b. linear (negative slope) c. quadratic d. cubic e. eponential f. logarithmic A. B. C. D. E. F. 8 SELECTING TOOLS To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. Eploring Gaussian and Logistic Models Work with a partner. Two common tpes of functions that are related to eponential functions are given. Use a graphing calculator to graph each function. Then determine the domain, range, intercept, and asmptote(s) of the function. 1 a. Gaussian Function: f () = e b. Logistic Function: f () = 1 + e Communicate Your Answer 3. How can ou recognize polnomial, eponential, and logarithmic models?. Use the Internet or some other reference to find real-life data that can be modeled using one of the tpes given in Eploration 1. Create a table and a scatter plot of the data. Then use a regression program to find a model that fits the data. Section 7.7 Modeling with Eponential and Logarithmic Functions 393
7.7 Lesson What You Will Learn Core Vocabular recursive rule for an eponential function, p. 39 Previous finite differences common ratio Classif data sets. Write eponential functions. Use technolog to find eponential and logarithmic models. Classifing Data You have analzed fi nite differences of data with equall-spaced inputs to determine what tpe of polnomial function can be used to model the data. For eponential data with equall-spaced inputs, the outputs are multiplied b a constant factor. So, consecutive outputs form a constant ratio. Classifing Data Sets Determine the tpe of function represented b each table. a. 1 0 1 3 0.5 1 8 1 3 b. 0 8 10 0 8 18 3 50 a. The inputs are equall spaced. Look for a pattern in the outputs. 1 0 1 3 0.5 1 8 1 3 As increases b 1, is multiplied b. So, the common ratio is, and the data in the table represent an eponential function. REMEMBER First differences of linear functions are constant, second differences of quadratic functions are constant, and so on. b. The inputs are equall spaced. The outputs do not have a common ratio. So, analze the finite differences. 0 8 10 0 8 18 3 50 10 1 18 first differences second differences The second differences are constant. So, the data in the table represent a quadratic function. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Determine the tpe of function represented b the table. Eplain our reasoning. 1. 0 10 0 30. 0 15 1 9 7 9 3 1 3 39 Chapter 7 Eponential and Logarithmic Functions
Writing Eponential Functions You know that two points determine a line. Similarl, two points determine an eponential curve. Writing an Eponential Function Using Two Points Write an eponential function = ab whose graph passes through (1, ) and (3, 5). REMEMBER You know that b must be positive b the definition of an eponential function. So, take the positive square root when solving for b. Step 1 Substitute the coordinates of the two given points into = ab. = ab 1 Equation 1: Substitute for and 1 for. 5 = ab 3 Equation : Substitute 5 for and 3 for. Step Solve for a in Equation 1 to obtain a = and substitute this epression for a b in Equation. 5 = ( b ) b3 Substitute for a in Equation. b 5 = b Simplif. 3 = b Solve for b. Step 3 Determine that a = b = 3 =. So, the eponential function is = (3 ). Data do not alwas show an eact eponential relationship. When the data in a scatter plot show an approimatel eponential relationship, ou can model the data with an eponential function. Finding an Eponential Model Year, Number of trampolines, 1 1 1 3 5 3 5 50 7 7 9 A store sells trampolines. The table shows the numbers of trampolines sold during the th ear that the store has been open. Write a function that models the data. Do ou think this model can be used to predict the number of trampolines sold in the 15th ear? Step 1 Make a scatter plot of the data. The data appear eponential. Step Choose an two points to write a model, such as (1, 1) and (, 3). Substitute the coordinates of these two points into = ab. 1 = ab 1 3 = ab Solve for a in the first equation to obtain a = 1 b. Substitute to obtain b = 3 3 1. and a = 1 8.3. 3 3 Trampoline Sales So, an eponential function that models the data is = 8.3(1.). The end behavior indicates that the model has unlimited growth as increases. While the model is valid for a few ears after the seventh ear, in the long run unlimited growth is not reasonable. Number of trampolines 80 0 0 0 0 0 Year Section 7.7 Modeling with Eponential and Logarithmic Functions 395
In real-life situations, ou can also show eponential relationships using recursive rules. REMEMBER Recall that for a sequence, a recursive rule gives the beginning term(s) of the sequence and a recursive equation that tells how a n is related to one or more preceding terms. A recursive rule for an eponential function gives the initial value of the function f (0), and a recursive equation that tells how a value f(n) is related to a preceding value f (n 1). Core Concept Writing Recursive Rules for Eponential Functions An eponential function of the form f () = ab is written using a recursive rule as follows. Recursive Rule f (0) = a, f (n) = r f (n 1) where a 0, r is the common ratio, and n is a natural number Eample = (3) can be written as f (0) =, f (n) = 3 f (n 1) initial value common ratio Notice that the base b of the eponential function is the common ratio r in the recursive equation. Also, notice the value of a in the eponential function is the initial value of the recursive rule. Writing a Recursive Rule for an Eponential Function STUDY TIP Notice that the domain consists of the natural numbers when written recursivel. Write a recursive rule for the function ou wrote in Eample 3. The function = 8.3(1.) is eponential with initial value f(0) = 8.3 and common ratio r = 1.. So, a recursive equation is f (n) = r f (n 1) Recursive equation for eponential functions = 1. f (n 1). Substitute 1. for r. A recursive rule for the eponential function is f (0) = 8.3, f (n) = 1. f (n 1). Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write an eponential function = ab whose graph passes through the given points. 3. (, 1), (3, ). (1, ), (3, 3) 5. (, 1), (5, ). WHAT IF? Repeat Eample 3 using the sales data from another store. Year, 1 3 5 7 Number of trampolines, 15 3 0 5 80 105 10 Write an recursive rule for the eponential function. 7. f () = (7) 8. f () = 9 ( 1 3 ) 39 Chapter 7 Eponential and Logarithmic Functions
Using Technolog You can use technolog to find best-fit models for eponential and logarithmic data. Finding an Eponential Model Use a graphing calculator to find an eponential model for the data in Eample 3. Then use each model to predict the number of trampolines sold in the eighth ear. Which prediction should ou use? Enter the data into a graphing calculator and perform an eponential regression. The model is = 8.(1.). Substitute = 8 into each model to predict the number of trampolines sold in the eighth ear. Eample 3: = 8.3(1.) 8 15 Regression model: = 8.(1.) 8 10 EpReg =a*b^ a=8.57377971 b=1.188803 All of the points were used to create the regression model, instead of onl two points as in Eample 3. So, use the prediction of 10 trampolines. Finding a Logarithmic Model The atmospheric pressure decreases with increasing altitude. At sea level, the average air pressure is 1 atmosphere (1.0337 kilograms per square centimeter). The table shows the pressures p (in atmospheres) at selected altitudes h (in kilometers). Use a graphing calculator to find a logarithmic model of the form h = a + b ln p that represents the data. Estimate the altitude when the pressure is 0.75 atmosphere. Air pressure, p 1 0.55 0.5 0.1 0.0 0.0 Altitude, h 0 5 10 15 0 5 Weather balloons carr instruments that send back information such as wind speed, temperature, and air pressure. Enter the data into a graphing calculator and perform a logarithmic regression. The model is h = 0.8.5 ln p. Substitute p = 0.75 into the model to obtain h = 0.8.5 ln 0.75.7. So, when the air pressure is 0.75 atmosphere, the altitude is about.7 kilometers. Monitoring Progress LnReg =a*bln a=.8578705 b=-.738985 r =.9955887 r=-.9971 Help in English and Spanish at BigIdeasMath.com 9. Use a graphing calculator to find an eponential model for the data in Monitoring Progress Question. 10. Use a graphing calculator to find a logarithmic model of the form p = a + b ln h for the data in Eample. Eplain wh the result is an error message. Section 7.7 Modeling with Eponential and Logarithmic Functions 397
7.7 Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE In the recursive rule for an eponential function f(0) = a, f(n) = r f(n 1), r is the and n is a.. WRITING Given a table of values, eplain how ou can determine whether an eponential function is a good model for a set of data pairs (, ). Monitoring Progress and Modeling with Mathematics In Eercises 3, determine the tpe of function represented b the table. Eplain our reasoning. (See Eample 1.) 3.. 0 3 9 1 15 0.5 1 1 5 3 1 0 1 1 8 1 1 1 ERROR ANALYSIS In Eercises 17 and 18, describe and correct the error in determining the tpe of function represented b the data. 17. 0 1 3 1 9 1 3 1 3 9 3 3 3 3 The outputs have a common ratio of 3, so the data represent a linear function. 5. 5 10 15 0 5 30 3 7 1 30 9 18. 1 1 3 1 8. 3 1 5 9 13 8 3 1 5 3 The outputs have a common ratio of, so the data represent an eponential function. In Eercises 7 1, write an eponential function = ab whose graph passes through the given points. (See Eample.) 7. (1, 3), (, 1) 8. (, ), (3, 1) 9. (3, 1), (5, ) 10. (3, 7), (5, 3) 11. (1, ), (3, 50) 1. (1, 0), (3, 0) 19. MODELING WITH MATHEMATICS A store sells motorized scooters. The table shows the numbers of scooters sold during the th ear that the store has been open. Write a function that models the data. Do ou think this model could be used to predict the number of motorized scooters sold after the 0th ear? (See Eample 3.) 13. ( 1, 10), (, 0.31) 1. (,.), (5, 09.) 15. 1. 1 ( 3, 10.8) (, ) 8 (, 3.) (1, 0.5) 1 9 1 3 19 5 5 37 53 7 71 398 Chapter 7 Eponential and Logarithmic Functions
0. MODELING WITH MATHEMATICS Your visual near point is the closest point at which our ees can see an object distinctl. The diagram shows the near point (in centimeters) at age (in ears). Write a function that models the data. Compare the average rate of change in the visual near point distances from age 0 to age 0 with that from age 0 to age 0. In Eercises 33 3, show that an eponential model fits the data. Then write a recursive rule that models the data. 33. n 0 1 3 5 f (n) 0.75 1.5 3 1 Visual Near Point Distances Age 0 1 cm Age 30 15 cm Age 0 5 cm 3. 35. n 0 1 3 f (n) 8 3 18 51 n 0 1 3 5 f (n) 9 8 1 3 Age 50 0 cm Age 0 100 cm In Eercises 1, determine whether the data show an eponential relationship. Then write a function that models the data. 1. 1 11 1 1 1 8 7 190 50 3. n 0 1 3 5 f (n) 1 5 18 3 37. USING EQUATIONS Complete a table of values for 0 n 5 using the given recursive rule of an eponential function. f (0) =, f (n) = 3 f (n 1) 38. USING STRUCTURE Write an eponential function for the recursive rule f (0) =, f (n) = 0.1 f (n 1). Eplain our reasoning.. 3.. 3 1 1 3 5 7 8 19 0 10 0 30 0 50 0 58 8 31 1 0 13 1 8 15 5 19 1 11 8 39. USING TOOLS Use a graphing calculator to find an eponential model for the data in Eercise 19. Then use the model to predict the number of motorized scooters sold in the tenth ear. (See Eample 5.) 0. USING TOOLS A doctor measures an astronaut s pulse rate (in beats per minute) at various times (in minutes) after the astronaut has finished eercising. The results are shown in the table. Use a graphing calculator to find an eponential model for the data. Then use the model to predict the astronaut s pulse rate after 1 minutes. In Eercises 5 3, write a recursive rule for the eponential function. (See Eample.) 5. f () = 3(7). f () = 5(9) 7. f () = 1(8) 8. f () = 19() 9. f () = 0.5(3) 30. f () = 1 3 () 31. f () = ( 1 ) 3. f () = 0.5 ( 3 ) 0 17 13 110 9 8 8 10 78 1 75 Section 7.7 Modeling with Eponential and Logarithmic Functions 399
1. USING TOOLS An object at a temperature of 10 C is removed from a furnace and placed in a room at 0 C. The table shows the temperatures d (in degrees Celsius) at selected times t (in hours) after the object was removed from the furnace. Use a graphing calculator to find a logarithmic model of the form t = a + b ln d that represents the data. Estimate how long it takes for the object to cool to 50 C. (See Eample.). HOW DO YOU SEE IT? Use the graph to write a recursive rule that models the data. f(n) 1 1 8 d 10 90 5 38 9 t 0 1 3 5 n. USING TOOLS The f-stops on a camera control the amount of light that enters the camera. Let s be a measure of the amount of light that strikes the film and let f be the f-stop. The table shows several f-stops on a 35-millimeter camera. Use a graphing calculator to find a logarithmic model of the form s = a + b ln f that represents the data. Estimate the amount of light that strikes the film when f = 5.57. f s 1.1 1.000.88 3.000 11.31 7 5. MAKING AN ARGUMENT Your friend sas it is possible to find a logarithmic model of the form d = a + b ln t for the data in Eercise 1. Is our friend correct? Eplain.. THOUGHT PROVOKING Is it possible to write as an eponential function of? Eplain our reasoning. (Assume p is positive.) 1 p p 3 p 8p 5 1p 3. DRAWING CONCLUSIONS The table shows the average weight (in kilograms) of an Atlantic cod that is ears old from the Gulf of Maine. Age, 1 3 5 Weight, 0.751 1.079 1.70.198 3.38 a. Find an eponential model for the data. Then write a recursive rule that models the data. b. B what percent does the weight of an Atlantic cod increase each ear in this period of time? Eplain. Maintaining Mathematical Proficienc Tell whether and are in a proportional relationship. Eplain our reasoning. (Skills Review Handbook) 8. = 9. = 3 1 50. = 5 7. CRITICAL THINKING You plant a sunflower seedling in our garden. The height h (in centimeters) of the seedling after t weeks can be modeled b the logistic function 5 h(t) = 1 + 13e 0.5t. a. Find the time it takes the sunflower seedling to reach a height of 00 centimeters. b. Use a graphing calculator to graph the function. Interpret the meaning of the asmptote in the contet of this situation. Reviewing what ou learned in previous grades and lessons 51. = Identif the focus, directri, and ais of smmetr of the parabola. Then graph the equation. (Section 3.3) 5. = 1 8 53. = 5. = 3 55. = 5 00 Chapter 7 Eponential and Logarithmic Functions
7.5 7.7 What Did You Learn? Core Vocabular eponential equations, p. 38 logarithmic equations, p. 387 recursive rule for an eponential function, p. 39 Core Concepts Section 7.5 Properties of Logarithms, p. 380 Change-of-Base Formula, p. 381 Section 7. Propert of Equalit for Eponential Equations, p. 38 Propert of Equalit for Logarithmic Equations, p. 387 Section 7.7 Classifing Data, p. 39 Writing Eponential Functions, p. 395 Writing Recursive Rules for Eponential Functions, p. 39 Using Eponential and Logarithmic Regression, p. 397 Mathematical Thinking 1. Eplain how ou used properties of logarithms to rewrite the function in part (b) of Eercise 5 on page 38.. How can ou use cases to analze the argument given in Eercise on page 391? Performance Task Measuring Natural Disasters In 005, an earthquake measuring.1 on the Richter scale barel shook the cit of Ocotillo, California, leaving virtuall no damage. But in 190, an earthquake with an estimated 8. on the same scale devastated the cit of San Francisco. Does twice the measurement on the Richter scale mean twice the intensit of the earthquake? To eplore the answer to this question and more, go to BigIdeasMath.com. 01
( 7 Chapter Review 7.1 Eponential Growth and Deca Functions (pp. 37 35) Tell whether the function = 3 represents eponential growth or eponential deca. Then graph the function. Step 1 Identif the value of the base. The base, 3, is greater than 1, so the function represents eponential growth. Step Make a table of values. 1 0 1 Step 3 Step Plot the points from the table. Draw, from left to right, a smooth curve that begins 1 9 1 3 1 3 9 just above the -ais, passes through the plotted points, and moves up to the right. Tell whether the function represents eponential growth or eponential deca. Identif the percent increase or decrease. Then graph the function. 1 ( 1, 3 1 (, 9 ( (, 9) 8 (1, 3) (0, 1) 1. f () = ( 1 3 ). = 5 3. f () = (0.). You deposit $1500 in an account that pas 7% annual interest. Find the balance after ears when the interest is compounded dail. 7. The Natural Base e (pp. 355 30) Simplif each epression. a. 18e13 e 7 = 9e13 7 = 9e b. (e 3 ) 3 = 3 (e 3 ) 3 = 8e 9 Simplif the epression. 5. e e 11. 0e 3 10e 7. ( 3e 5 ) Tell whether the function represents eponential growth or eponential deca. Then graph the function. 8. f () = 1 3 e 9. = e 10. = 3e 0.75 7.3 Logarithms and Logarithmic Functions (pp. 31 38) Find the inverse of the function f () = ln( ). = ln( ) Set equal to f (). = ln( ) Switch and. e = Write in eponential form. e + = Add to each side. The inverse of f () = ln( ) is f 1 () = e +. Check 9 9 The graphs appear to be reflections of each other in the line =. 0 Chapter 7 Eponential and Logarithmic Functions
Rewrite the equation in eponential or logarithmic form. Then find the value of. 11. = 8 1. log 3 = 13. log 5 = 3 Find the inverse of the function. 1. f () = 8 15. f () = ln( ) 1. f () = log( + 9) 17. Graph f () = log 1/5. Identif the domain and range of the function. 7. Transformations of Eponential and Logarithmic Functions (pp. 39 37) Describe the transformation of f () = ( 1 g() = ( 1 3) 1 3) represented b + 3. Then graph each function. Notice that the function is of the form g() = ( 1 3) h + k, where h = 1 and k = 3. f g So, the graph of g is a translation 1 unit right and 3 units up of the graph of f. Describe the transformation of f represented b g. Then graph each function. 18. f () = e, g() = e 5 8 19. f () = log, g() = 1 log ( + 5) Write a rule for g. 0. Let the graph of g be a vertical stretch b a factor of 3, followed b a translation units left and 3 units up of the graph of f () = e. 1. Let the graph of g be a translation units down, followed b a reflection in the -ais of the graph of f () = log. 7.5 Properties of Logarithms (pp. 379 38) Epand ln 15. ln 15 = ln 1 5 ln Quotient Propert = ln 1 + ln 5 ln Product Propert = ln 1 + 5 ln ln Power Propert Epand or condense the logarithmic epression.. log 8 3 3. log 10 3. ln 3 5 5. 3 log 7 + log 7. log 1 log 7. ln + 5 ln ln 8 Use the change-of-base formula to evaluate the logarithm. 8. log 10 9. log 7 9 30. log 3 Chapter 7 Chapter Review 03
7. Solving Eponential and Logarithmic Equations (pp. 385 39) Solve ln(3 9) = ln( + ). ln(3 9) = ln( + ) 3 9 = + 9 = = 15 Write original equation. Propert of Equalit for Logarithmic Equations Subtract from each side. Add 9 to each side. Check ln(3 15 9) =? ln( 15 + ) ln(5 9) =? ln(30 + ) ln 3 = ln 3 Solve the equation. Check for etraneous solutions. 31. 5 = 8 3. log 3 ( 5) = 33. ln + ln( + ) = 3 Solve the inequalit. 3. > 1 35. ln 9 3. e 1 7.7 Modeling with Eponential and Logarithmic Functions (pp. 393 00) Write an eponential function whose graph passes through (1, 3) and (, ). Step 1 Substitute the coordinates of the two given points into = ab. 3 = ab 1 Equation 1: Substitute 3 for and 1 for. = ab Equation : Substitute for and for. Step Solve for a in Equation 1 to obtain a = 3 and substitute this epression for a in Equation. b = ( 3 b ) b Substitute 3 for a in Equation. b = 3b 3 Simplif. 8 = b 3 Divide each side b 3. = b Take cube root of each side. Step 3 Determine that a = 3 b = 3. So, the eponential function is = 3 ( ). Write an eponential model for the data pairs (, ). 37. (3, 8), (5, ) 38. 1 3 5.1 7.39 10.59 15.18 39. A shoe store sells a new tpe of basketball shoe. The table shows the pairs sold s over time t (in weeks). Use a graphing calculator to find a logarithmic model of the form s = a + b ln t that represents the data. Estimate how man pairs of shoes are sold after weeks. Week, t 1 3 5 7 9 Pairs sold, s 5 3 8 58 5 0 Chapter 7 Eponential and Logarithmic Functions