Exponential and Logarithmic Functions

Transcription

1 7 Eponential and Logarithmic Functions 7. Graph Eponential Growth Functions 7.2 Graph Eponential Deca Functions 7.3 Use Functions Involving e 7.4 Evaluate Logarithms and Graph Logarithmic Functions 7.5 Appl Properties of Logarithms 7.6 Solve Eponential and Logarithmic Equations 7.7 Write and Appl Eponential and Power Functions Before In previous chapters, ou learned the following skills, which ou ll use in Chapter 7: graphing functions, finding inverse functions, and writing functions. Prerequisite Skills VOCABULARY CHECK Cop and complete the statement using the graph at the right.. The domain of the function is?. 2. The range of the function is?. 3. The inverse of the function is? SKILLS CHECK Graph the function. State the domain and range. (Review p. 446 for ) Ï Ï Ï Find the inverse of the function. (Review p. 438 for 7.4.) , Write a quadratic function in standard form for the parabola that passes through the given points. (Review p. 309 for 7.7.) 0. (0, 2), (, 2), (3, 4). (3, 8), (4, 7), (7, 56) 2. (23, 9), (, 27), (5, 255) SFSFRVJTJUF TLJMMT QSBDUJDF BU DMBTT[POF DPN Take-Home Tutor for problem solving help at n2pe-0700.indd 476 0/4/05 2:20:34 PM

2 Now In Chapter 7, ou will appl the big ideas listed below and reviewed in the Chapter Summar on page 538. You will also use the ke vocabular listed below. Big Ideas Graphing eponential and logarithmic functions 2 Solving eponential and logarithmic equations 3 Writing and appling eponential and power functions KEY VOCABULARY eponential function, p. 478 eponential growth function, p. 478 growth factor, p. 478 asmptote, p. 478 eponential deca function, p. 486 deca factor, p. 486 natural base e, p. 492 logarithm of with base b, p. 499 common logarithm, p. 500 natural logarithm, p. 500 eponential equation, p. 55 logarithmic equation, p. 57 Wh? You can use eponential and logarithmic functions to model man scientific relationships. For eample, ou can use a logarithmic function to relate the size of a telescope lens and the abilit of the telescope to see certain stars. Algebra The animation illustrated below for Eample 7 on page 59 helps ou answer this question: How is the diameter of a telescope s objective lens related to the apparent magnitude of the dimmest star that can be seen with the telescope? The magnitude of stars is a measure of their brightness as viewed from Earth. Solve to find the diameter of a telescope that reveals stars of a given magnitude. Algebra at classzone.com Other animations for Chapter 7: pages 480, 487, 502, and

3 7. Graph Eponential Growth Functions Before EpoBefNowWh You graphed polnomial uergiu ;kjer and ;kejrng radical functions. ;kjer ;er e;rg ;erg erg ewrkj. Now BNWtet You will graph sfig ;lsdfgj and use sdkjfdfgkjs eponential dfg growth wtireuh functions. d cv;kaurg erbg serg. Wh? BNWtet So ou can sdlkgfj model sd;lkjgf sports sdljgf equipment s;jgf sdfg costs, sdfg as js;dfg in E. js;dlgf 40. pijg f; nsdfn sdfgdhf. Ke Vocabular eponential function eponential growth function growth factor asmptote An eponential function has the form 5 ab where a Þ 0 and the base b is a positive number other than. If a > 0 and b >, then the function 5 ab is an eponential growth function, and b is called the growth factor. The simplest tpe of eponential growth function has the form 5 b. KEY CONCEPT Parent Function for Eponential Growth Functions For Your Notebook The function f() 5 b, where b >, is the parent function for the famil of eponential growth functions with base b. The general shape of the graph of f() 5 b is shown below. f () 5 b (b. ) The -ais is an asmptote of the graph. An asmptote is a line that a graph approaches more and more closel. (0, ) (, b) The graph rises from left to right, passing through the points (0, ) and (, b). The domain of f() 5 b is all real numbers. The range is > 0. E XAMPLE Graph 5 b for b > Graph 5 2. Solution STEP STEP 2 STEP 3 Make a table of values Plot the points from the table. Draw, from left to right, a smooth curve that begins just above the -ais, passes through the plotted points, and moves up to the right. 4 s2, 2 d s22, 4 d (2, 4) (, 2) (0, ) (3, 8) Chapter 7 Eponential and Logarithmic Functions

4 The graph of a function 5 ab is a vertical stretch or shrink of the graph of 5 b. The -intercept of the graph of 5 ab occurs at (0, a) rather than (0, ). E XAMPLE 2 Graph 5 ab for b > Graph the function. a. 5 2 p 4 b Solution a. Plot 0, 2 2 and (, 2). Then, from left to right, draw a curve that begins just above the -ais, passes through the two points, and moves up to the right. b. Plot (0, 2) and, Then, from left to right, draw a curve that begins just below the -ais, passes through the two points, and moves down to the right. CLASSIFY FUNCTIONS Note that the function in part (b) of Eample 2 is not an eponential growth function because a 5 2 < 0. 3 s0, 2 d 5? 4 2 (0, 2) s, 2 2d (, 2) c TRANSLATIONS To graph a function of the form 5 ab 2 h k, begin b sketching the graph of 5 ab. Then translate the graph horizontall b h units and verticall b k units. E XAMPLE 3 Graph 5 ab 2 h k for b > Graph 5 4 p State the domain and range. Solution Begin b sketching the graph of 5 4 p 2, which passes through (0, 4) and (, 8). Then translate the graph right unit and down 3 units to obtain the graph of 5 4 p (, 8) (0, 4) (2, 5) The graph s asmptote is the line The domain is all real numbers, and the range is > ? 2 (, ) 5 4? GUIDED PRACTICE for Eamples, 2, and 3 Graph the function. State the domain and range p 3 3. f() Graph Eponential Growth Functions 479

5 EXPONENTIAL GROWTH MODELS When a real-life quantit increases b a fied percent each ear (or other time period), the amount of the quantit after t ears can be modeled b the equation 5 a( r) t where a is the initial amount and r is the percent increase epressed as a decimal. Note that the quantit r is the growth factor. E XAMPLE 4 Solve a multi-step problem COMPUTERS In 996, there were 2573 computer viruses and other computer securit incidents. During the net 7 ears, the number of incidents increased b about 92% each ear. Write an eponential growth model giving the number n of incidents t ears after 996. About how man incidents were there in 2003? Graph the model. Use the graph to estimate the ear when there were about 25,000 computer securit incidents. Virus Alert! A virus has been detected! Virus name: snake File name: essa.doc Action: Virus was cleaned from file File essa.doc might be damaged! OK More Info Solution STEP The initial amount is a and the percent increase is r So, the eponential growth model is: n 5 a( r) t Write eponential growth model. AVOID ERRORS Notice that the percent increase and the growth factor are two different values. An increase of 92% corresponds to a growth factor of.92. STEP 2 STEP ( 0.92) t Substitute 2573 for a and 0.92 for r (.92) t Simplif. Using this model, ou can estimate the number of incidents in 2003 (t 5 7) to be n (.92) 7 ø 247,485. The graph passes through the points (0, 2573) and (, ). Plot a few other points. Then draw a smooth curve through the points. Using the graph, ou can estimate that the number of incidents was about 25,000 during 2002 (t ø 6). at classzone.com Number of incidents n 250, ,000 50,000 00,000 50, t Years since 996 GUIDED PRACTICE for Eample 4 4. WHAT IF? In Eample 4, estimate the ear in which there were about 250,000 computer securit incidents. 5. In the eponential growth model 5 527(.39), identif the initial amount, the growth factor, and the percent increase. 480 Chapter 7 Eponential and Logarithmic Functions

6 COMPOUND INTEREST Eponential growth functions are used in real-life situations involving compound interest. Compound interest is interest paid on the initial investment, called the principal, and on previousl earned interest. Interest paid onl on the principal is called simple interest. KEY CONCEPT For Your Notebook 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 this equation: A 5 P n r 2 nt E XAMPLE 5 Find the balance in an account FINANCE You deposit $4000 in an account that pas 2.92% annual interest. Find the balance after ear if the interest is compounded with the given frequenc. a. Quarterl b. Dail Solution a. With interest compounded quarterl, the balance after ear is: A 5 P r n 2 nt Write compound interest formula p P , r , n 5 4, t (.0073) 4 Simplif. ø Use a calculator. c The balance at the end of ear is $ b. With interest compounded dail, the balance after ear is: A 5 P r n 2 nt Write compound interest formula p P , r , n 5 365, t (.00008) 365 Simplif. ø Use a calculator. c The balance at the end of ear is $ GUIDED PRACTICE for Eample 5 6. FINANCE You deposit $2000 in an account that pas 4% annual interest. Find the balance after 3 ears if the interest is compounded dail. 7. Graph Eponential Growth Functions 48

7 7. EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 7, 29, and 37 5 STANDARDIZED TEST PRACTICE Es. 2, 24, 25, 32, 40, and 4 5 MULTIPLE REPRESENTATIONS E. 42. VOCABULARY In the eponential growth model 5 2.4(.5), identif the initial amount, the growth factor, and the percent increase. 2. WRITING What is an asmptote? EXAMPLES and 2 on pp for Es. 3 4 MATCHING GRAPHS Match the function with its graph p p p 3 A. B. 2 (0, 23) 3 (, 26) C. (, 6) (0, 2) (, 6) (0, 3) GRAPHING FUNCTIONS Graph the function f() 5 5 p p 4. g() 5 2(.5) p 3 4. h() 5 22(2.5) EXAMPLE 3 on p. 479 for Es TRANSLATING GRAPHS Graph the function. State the domain and range p p p p f() 5 6 p g() 5 5 p h() 5 22 p MULTIPLE CHOICE The graph of which function is shown? A f() 5 2(.5) 2 B f() 5 2(.5) C f() 5 3(.5) 2 D f() 5 3(.5) (0, 3) (, 4) 25. MULTIPLE CHOICE The student enrollment E of a high school was 30 in 998 and has increased b 0% per ear since then. Which eponential growth model gives the school s student enrollment in terms of t, where t is the number of ears since 998? A E 5 0.(30) t C E 5.(30) t B E 5 30(0.) t D E 5 30(.) t 482 Chapter 7 Eponential and Logarithmic Functions

8 ERROR ANALYSIS Describe and correct the error in graphing the function p (, 8) (, 7) ( 2, 5) 2 (0, ) WRITING MODELS In Eercises 28 30, write an eponential growth model that describes the situation. 28. In 992, 29 monk parakeets were observed in the United States. For the net ears, about 2% more parakeets were observed each ear. 29. You deposit $800 in an account that pas 2% annual interest compounded dail. 30. You purchase an antique table for $450. The value of the table increases b 6% per ear. 3. GRAPHING CALCULATOR You deposit $500 in a bank account that pas 3% annual interest compounded earl. a. Tpe 500 into a graphing calculator and press. Then enter the formula ANS *.03, as shown at the right. Press seven times to find our balance after 7 ears. b. Find the number of ears it takes for our balance to eceed $ OPEN-ENDED MATH Write an eponential function of the form 5 ab 2 h k whose graph has a -intercept of 5 and an asmptote of GRAPHING CALCULATOR Consider the eponential growth function 5 ab 2 h k where a 5 2, b 5 5, h 5 24, and k 5 3. Predict the effect on the function s graph of each change in a, b, h, or k described in parts (a) (d). Use a graphing calculator to check our prediction. a. a changes to b. b changes to 4 c. h changes to 3 d. k changes to CHALLENGE Consider the eponential function f() 5 ab. f ( ) a. Show that 5 b. f () b. Use the result from part (a) to eplain wh there is no eponential function of the form f() 5 ab whose graph passes through the points in the table below Ans* Graph Eponential Growth Functions 483

9 PROBLEM SOLVING EXAMPLE 4 on p. 480 for Es DVD PLAYERS From 997 to 2002, the number n (in millions) of DVD plaers sold in the United States can be modeled b n (2.47) t where t is the number of ears since 997. a. Identif the initial amount, the growth factor, and the annual percent increase. b. Graph the function. Estimate the number of DVD plaers sold in INTERNET Each March from 998 to 2003, a website recorded the number of referrals it received from Internet search engines. The results can be modeled b (.50) t where t is the number of ears since 998. a. Identif the initial amount, the growth factor, and the annual percent increase. b. Graph the function and state the domain and range. Estimate the number of referrals the website received from Internet search engines in March of EXAMPLE 5 on p. 48 for Es ACCOUNT BALANCE You deposit $2200 in a bank account. Find the balance after 4 ears for each of the situations described below. a. The account pas 3% annual interest compounded quarterl. b. The account pas 2.25% annual interest compounded monthl. c. The account pas 2% annual interest compounded dail. 38. DEPOSITING FUNDS You want to have $3000 in our savings account after 3 ears. Find the amount ou should deposit for each of the situations described below. a. The account pas 2.25% annual interest compounded quarterl. b. The account pas 3.5% annual interest compounded monthl. c. The account pas 4% annual interest compounded earl. 39. MULTI-STEP PROBLEM In 990, the population of Austin, Teas, was 494,290. During the net 0 ears, the population increased b about 3% each ear. a. Write a model giving the population P (in thousands) of Austin t ears after 990. What was the population in 2000? b. Graph the model and state the domain and range. c. Estimate the ear when the population was about 590,000. Austin, Teas 40. SHORT RESPONSE At an online auction, the opening bid for a pair of in-line skates is $50. The price of the skates increases b 0.5% per bid during the net 6 bids. a. Write a model giving the price p (in dollars) of the skates after n bids. b. What was the price after 5 bids? According to the model, what will the price be after 00 bids? Is this predicted price reasonable? Eplain. 5 WORKED-OUT SOLUTIONS 484 Chapter 7 Eponential p. WS and Logarithmic Functions 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

10 4. EXTENDED RESPONSE In 2000, the average price of a football ticket for a Minnesota Viking s game was $ During the net 4 ears, the price increased an average of 6% each ear. a. Write a model giving the average price p (in dollars) of a ticket t ears after b. Graph the model. Estimate the ear when the average price of a ticket was about $60. c. Eplain how ou can use the graph of p(t) to determine the minimum and maimum t-values in the domain for which the function gives meaningful results. 42. MULTIPLE REPRESENTATIONS In 977, there were 4 breeding pairs of bald eagles in Marland. Over the net 24 ears, the number of breeding pairs increased b about 8.9% each ear. a. Writing an Equation Write a model giving the number n of breeding pairs of bald eagles in Marland t ears after 977. b. Making a Table Make a table of values for the model. c. Drawing a Graph Graph the model. d. Using a Graph About how man breeding pairs of bald eagles were in Marland in 200? 43. REASONING Is investing $3000 at 6% annual interest and $3000 at 8% annual interest equivalent to investing $6000 (the total of the two principals) at 7% annual interest (the average of the two interest rates)? Eplain. 44. CHALLENGE The earl cost for residents to attend a state universit has increased from $5200 to $9000 in the last 5 ears. a. To the nearest tenth of a percent, what has been the average annual growth rate in cost? b. If this growth rate continues, what will the cost be in 5 more ears? MIXED REVIEW PREVIEW Prepare for Lesson 7.2 in Es Evaluate the power. 45. (0.6) 3 (p. 0) 46. (0.4) 2 (p. 0) 47. (0.5) 5 (p. 0) 48. (0.25) 3 (p. 0) (p. 330) (p. 330) (p. 330) (p. 330) Factor the epression (p. 252) (p. 252) (p. 259) (p. 259) (p. 353) (p. 353) Solve the equation. (p. 44) ( 2) ( 2 9) EXTRA PRACTICE for Lesson 7., p ONLINE Graph Eponential QUIZ at classzone.com Growth Functions 485

11 7.2 Graph Eponential Deca Functions Before You graphed and used eponential growth functions. Now You will graph and use eponential deca functions. Wh? So ou can model depreciation, as in E. 3. Ke Vocabular eponential deca function deca factor In Lesson 7. ou studied eponential growth functions. In this lesson, ou will stud eponential deca functions, which have the form 5 ab where a > 0 and 0 < b <. The base b of an eponential deca function is called the deca factor. KEY CONCEPT For Your Notebook Parent Function for Eponential Deca Functions The function f() 5 b, where 0 < b <, is the parent function for the famil of eponential deca functions with base b. The general shape of the graph of f() 5 b is shown below. The graph falls from left to right, passing through the points (0, ) and (, b). f () 5 b (0, b, ) (0, ) (, b) The -ais is an asmptote of the graph. The domain of f() 5 b is all real numbers. The range is > 0. E XAMPLE Graph 5 b for 0 < b < Graph Solution STEP Make a table of values STEP 2 Plot the points from the table. STEP 3 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. 4 (23, 8) (22, 4) 3 (2, 2) (0, ) 2 5 s d s, 2d s2, 4d 486 Chapter 7 Eponential and Logarithmic Functions

12 TRANSFORMATIONS Recall from Lesson 7. that the graph of a function 5 ab is a vertical stretch or shrink of the graph of 5 b, and the graph of 5 ab 2 h k is a translation of the graph of 5 ab. E XAMPLE 2 Graph 5 ab for 0 < b < CLASSIFY FUNCTIONS Note that the function in part (b) of Eample 2 is not an eponential deca function because a 5 23 < 0. Graph the function. a b Solution a. Plot (0, 2) and, right to left, draw a curve that begins just above the -ais, passes through the two points, and moves up to the left Then, from right to left, draw a curve that begins just below the -ais, passes through the two points, and moves down to the left Then, from b. Plot (0, 23) and, 2 6 (0, 2) s, 2d c (0, 23) 6 s, 2 5d c at classzone.com GUIDED PRACTICE for Eamples and 2 Graph the function f() E XAMPLE 3 Graph 5 ab 2 h k for 0 < b < Graph State the domain and range. Solution Begin b sketching the graph of , which passes through (0, 3) and, Then translate the graph left unit and down 2 units. Notice that the translated graph passes through (2, ) and 0, The graph s asmptote is the line The domain is all real numbers, and the range is > (2, ) s0, 2 2d (0, 3) 3 s, 2d c c Graph Eponential Deca Functions 487

13 EXPONENTIAL DECAY MODELS When a real-life quantit decreases b a fied percent each ear (or other time period), the amount of the quantit after t ears can be modeled b the equation 5 a( 2 r) t where a is the initial amount and r is the percent decrease epressed as a decimal. Note that the quantit 2 r is the deca factor. E XAMPLE 4 Solve a multi-step problem SNOWMOBILES A new snowmobile costs $4200. The value of the snowmobile decreases b 0% each ear. Write an eponential deca model giving the snowmobile s value (in dollars) after t ears. Estimate the value after 3 ears. Graph the model. Use the graph to estimate when the value of the snowmobile will be $2500. AVOID ERRORS Notice that the percent decrease, 0%, tells ou how much value the snowmobile loses each ear. The deca factor, 0.90, tells ou what fraction of the snowmobile s value remains each ear. Solution STEP The initial amount is a and the precent decrease is r So, the eponential deca model is: STEP 2 STEP 3 5 a( 2 r) t Write eponential deca model ( 2 0.0) t Substitute 4200 for a and 0.0 for r (0.90) t Simplif. When t 5 3, the snowmobile s value is (0.90) 3 5 $ The graph passes through the points (0, 4200) and (, 3780). It has the t-ais as an asmptote. Plot a few other points. Then draw a smooth curve through the points. Using the graph, ou can estimate that the value of the snowmobile will be $2500 after about 5 ears. Value (dollars) t Years GUIDED PRACTICE for Eamples 3 and 4 Graph the function. State the domain and range g() WHAT IF? In Eample 4, suppose the value of the snowmobile decreases b 20% each ear. Write and graph an equation to model this situation. Use the graph to estimate when the value of the snowmobile will be $ SNOWMOBILE The value of a snowmobile has been decreasing b 7% each ear since it was new. After 3 ears, the value is $3000. Find the original cost of the snowmobile. 488 Chapter 7 Eponential and Logarithmic Functions

14 7.2 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 9, 9, and 33 5 STANDARDIZED TEST PRACTICE Es. 2, 5, 27, 28, 33, and 35. VOCABULARY In the eponential deca model 5 250(0.85) t, identif the initial amount, the deca factor, and the percent decrease. 2. WRITING Eplain how to tell whether the function 5 b represents eponential growth or eponential deca. CLASSIFYING FUNCTIONS Tell whether the function represents eponential growth or eponential deca. 3. f() f() f() p 4 6. f() 5 25(0.25) EXAMPLES and 2 on pp for Es. 7 5 GRAPHING FUNCTIONS Graph the function f() (0.2) g() 5 2(0.75) h() MULTIPLE CHOICE The graph of which function is shown? A B C D (0, 22) 25 s 6, 2 5d EXAMPLE 3 on p. 487 for Es TRANSLATING GRAPHS Graph the function. State the domain and range (0.25) f() g() h() GRAPHING CALCULATOR Consider the eponential deca function 5 ab 2 h k where a 5 3, b 5 0.4, h 5 2, and k 5 2. Predict the effect on the function s graph of each change in a, b, h, or k described in parts (a) (d). Use a graphing calculator to check our prediction. a. a changes to 4 b. b changes to 0.2 c. h changes to 5 d. k changes to ERROR ANALYSIS You invest $500 in the stock of a compan. The value of the stock decreases 2% each ear. Describe and correct the error in writing a model for the value of the stock after t ears. 5 ( amount)( Initial factor) Deca t 5 500(0.02) t 7.2 Graph Eponential Deca Functions 489

15 27. MULTIPLE CHOICE What is the asmptote of the graph of ? A 523 B 522 C 5 2 D OPEN-ENDED MATH Write an eponential function whose graph lies between the graphs of 5 (0.5) and 5 (0.25) CHALLENGE Do f() 5 5(4) 2 and g() 5 5(0.25) represent the same function? Justif our answer. PROBLEM SOLVING EXAMPLE 4 on p. 488 for Es MEDICINE When a person takes a dosage of I milligrams of ibuprofen, the amount A (in milligrams) of medication remaining in the person s bloodstream after t hours can be modeled b the equation A 5 I(0.7) t. Amount of Ibuprofen in Bloodstream Medication (mg) A I 0.7I 0.50I 0.36I A 5 I(0.7) t t 5 0 t 5 t t Time (hours) t 5 3 Find the amount of ibuprofen remaining in a person s bloodstream for the given dosage and elapsed time since the medication was taken. a. Dosage: 200 mg b. Dosage: 325 mg c. Dosage: 400 mg Time:.5 hours Time: 3.5 hours Time: 5 hours 3. BIKE COSTS You bu a new mountain bike for $200. The value of the bike decreases b 25% each ear. a. Write a model giving the mountain bike s value (in dollars) after t ears. Use the model to estimate the value of the bike after 3 ears. b. Graph the model. c. Estimate when the value of the bike will be $ DEPRECIATION The table shows the amount d that a boat depreciates during each ear t since it was new. Show that the ratio of depreciation amounts for consecutive ears is constant. Then write an equation that gives d as a function of t. Year, t Depreciation, d $906 $832 $762 $692 $ WORKED-OUT SOLUTIONS on p. WS 5 STANDARDIZED TEST PRACTICE

16 33. SHORT RESPONSE The value of a car can be modeled b the equation 5 24,000(0.845) t where t is the number of ears since the car was purchased. a. Graph the model. Estimate when the value of the car will be $0,000. b. Use the model to predict the value of the car after 50 ears. Is this a reasonable value? Eplain. 34. MULTI-STEP PROBLEM When a plant or animal dies, it stops acquiring carbon-4 from the atmosphere. Carbon-4 decas over time with a half-life of about 5730 ears. The percent P of the original amount of carbon-4 that remains in a sample after t ears is given b this equation: P t/5730 a. What percent of the original carbon-4 remains in a sample after 2500 ears? 5000 ears? 0,000 ears? b. Graph the model. c. An archaeologist found a bison bone that contained about 37% of the carbon-4 present when the bison died. Use the graph to estimate the age of the bone when it was found. 35. EXTENDED RESPONSE The number E of eggs a Leghorn chicken produces per ear can be modeled b the equation E (0.89) w/52 where w is the age (in weeks) of the chicken and w 22. a. Interpret Identif the deca factor and the percent decrease. b. Graph Graph the model. c. Estimate Estimate the egg production of a chicken that is 2.5 ears old. d. Reasoning Eplain how ou can rewrite the given equation so that time is measured in ears rather than in weeks. 36. CHALLENGE You bu a new stereo for $300 and are able to sell it 4 ears later for $275. Assume that the resale value of the stereo decas eponentiall with time. Write an equation giving the stereo s resale value V (in dollars) as a function of the time t (in ears) since ou bought it. MIXED REVIEW PREVIEW Prepare for Lesson 7.3 in Es Graph the function. State the domain and range (p. 337) (p. 337) (p. 337) Ï 3 (p. 446) 4. 5 Ï 2 4 (p. 446) Ï 2 (p. 446) p 2 (p. 478) (p. 478) p 5 (p. 478) Verif that f and g are inverse functions. (p. 438) 46. f() , g() f() , g() f() , g() /3 49. f() , g() 5 Ï EXTRA PRACTICE for Lesson 7.2, p ONLINE Graph Eponential QUIZ at classzone.com Deca Functions 49

17 7.3 Use Functions Involving e Before You studied eponential growth and deca functions. Now You will stud functions involving the natural base e. Wh? So ou can model visibilit underwater, as in E. 59. Ke Vocabular 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 ( ). The epression n 2 n approaches e as n increases. n n n KEY CONCEPT For Your Notebook The Natural Base e The natural base e is irrational. It is defined as follows: As n approaches `, n 2 n approaches e ø E XAMPLE Simplif natural base epressions REVIEW EXPONENTS For help with properties of eponents, see p Simplif the epression. a. e 2 p e 5 5 e 2 5 b. 2e 4 3e 3 5 4e4 2 3 c. (5e 23 ) (e 23 ) 2 5 e 7 5 4e 5 25e e 6 E XAMPLE 2 Evaluate natural base epressions Use a calculator to evaluate the epression. Epression Kestrokes Displa a. e 4 [e ] b. e [e ] Chapter 7 Eponential and Logarithmic Functions

18 GUIDED PRACTICE for Eamples and 2 Simplif the epression.. e 7 p e e 23 p 6e Use a calculator to evaluate e 3/4. 24e 8 4e 5 4. (0e 24 ) 3 KEY CONCEPT For Your Notebook Natural Base Functions A function of the form 5 ae r is called a natural base eponential function. If a > 0 and r > 0, the function is an eponential growth function. If a > 0 and r < 0, the function is an eponential deca function. The graphs of the basic functions 5 e and 5 e 2 are shown below. Eponential growth 5 e 5 e 2 Eponential deca 3 (0, ) (, 2.78) 3 (0, ) (, 0.368) E XAMPLE 3 Graph natural base functions Graph the function. State the domain and range. a. 5 3e 0.25 b. 5 e 20.75( 2 2) ANOTHER WAY You can also write the function from part (a) in the form 5 ab in order to graph it: 5 3e (e 0.25 ) ø 3(.28) Solution a. Because a 5 3 is positive and b. a 5 is positive and r r is positive, the function is is negative, so the function is an eponential growth function. an eponential deca function. Plot the points (0, 3) and (, 3.85) Translate the graph of 5 e and draw the curve. right 2 units and up unit. (, 3.85) (0, 3) 5 3e 0.25 (22, 4.48) 5 e (0, ) 5 e 20.75( 2 2) (0, 5.48) (2, 2) The domain is all real numbers, The domain is all real numbers, and the range is > 0. and the range is >. 7.3 Use Functions Involving e 493

19 E XAMPLE 4 Solve a multi-step problem BIOLOGY The length l (in centimeters) of a tiger shark can be modeled b the function Adult shark l e 20.78t where t is the shark s age (in ears). Graph the model. Use the graph to estimate the length of a tiger shark that is 3 ears old. Newborn shark INTERPRET VARIABLES On a graphing calculator, enter the function l e 20.78t using the variables and, as shown below: e Solution STEP STEP 2 Graph the model, as shown. Use the trace feature to determine that l ø 75 when t 5 3. c The length of a 3-ear-old tiger shark is about 75 centimeters. X=3 Y= GUIDED PRACTICE for Eamples 3 and 4 Graph the function. State the domain and range e f() 5 2 e e 0.25( 2 ) WHAT IF? In Eample 4, use the given function to estimate the length of a tiger shark that is 5 ears old. CONTINUOUSLY COMPOUNDED INTEREST In Lesson 7., ou learned that the balance of an account earning compound interest is given b this formula: A 5 P n r 2 nt As the frequenc n of compounding approaches positive infinit, the compound interest formula approimates the following formula. KEY CONCEPT For Your Notebook Continuousl Compounded Interest When interest is compounded continuousl, the amount A in an account after t ears is given b the formula A 5 Pe rt where P is the principal and r is the annual interest rate epressed as a decimal. 494 Chapter 7 Eponential and Logarithmic Functions

20 E XAMPLE 5 Model continuousl compounded interest FINANCE You deposit $4000 in an account that pas 6% annual interest compounded continuousl. What is the balance after ear? Solution Use the formula for continuousl compounded interest. A 5 Pe rt Write formula e 0.06() Substitute 4000 for P, 0.06 for r, and for t. ø Use a calculator. c The balance at the end of ear is $ GUIDED PRACTICE for Eample 5 0. FINANCE You deposit $2500 in an account that pas 5% annual interest compounded continuousl. Find the balance after each amount of time. a. 2 ears b. 5 ears c. 7.5 ears. FINANCE Find the amount of interest earned in parts (a) (c) of Eercise EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5, 35, and 57 5 STANDARDIZED TEST PRACTICE Es. 2, 5, 6, 52, 53, and 60. VOCABULARY Cop and complete: The number? is an irrational number approimatel equal to WRITING Tell whether the function f() 5 3 e4 is an eample of eponential growth or eponential deca. Eplain. EXAMPLE on p. 492 for Es. 3 8 SIMPLIFYING EXPRESSIONS Simplif the epression. 3. e 3 p e 4 4. e 22 p e 6 5. (2e 3 ) 3 6. (2e 22 ) (3e 5 ) 2 8. e p e 23 p e 4 9. Ï 9e 6 0. e p 5e 3. 3e e 2. 4e e Ï 8e e 4 8e 5. MULTIPLE CHOICE What is the simplified form of (4e 2 ) 3? A 4e 6 B 4e 8 C 64e 6 D 64e 8 6. MULTIPLE CHOICE What is the simplified form of Î 4(27e 3 ) 3e 7 23? A 6e 0 B 6e 6 4 C 6e 3 2 D 6e Use Functions Involving e 495

21 ERROR ANALYSIS Describe and correct the error in simplifing the epression. 7. (3e 5 ) 2 5 3e (5)(2) 5 3e 0 8. e e6 22 e 5 e 4 EXAMPLE 2 on p. 492 for Es EVALUATING EXPRESSIONS Use a calculator to evaluate the epression. 9. e e 23/4 2. e e /2 23. e 22/5 24. e e e e e 2/ e e 4. GROWTH OR DECAY Tell whether the function is an eample of eponential growth or eponential deca. 3. f() 5 3e f() 5 3 e4 33. f() 5 e f() e 35. f() 5 4 e f() 5 e f() 5 2e f() 5 4e 22 EXAMPLE 3 on p. 493 for Es MATCHING GRAPHS Match the function with its graph e e e A. B. C. (22, 2.37) (0, 3) 3 (0, 0.5) (, 0.82) 3 (0, 2) (, 3.30) GRAPHING FUNCTIONS Graph the function. State the domain and range e e e e e e f() 5 2 e g() e h() 5 e 22( ) GRAPHING CALCULATOR Use the table feature of a graphing calculator to find the value of n for which n 2 n gives the value of e correct to 9 decimal places. Eplain the process ou used to find our answer. 52. SHORT RESPONSE Can e be epressed as a ratio of two integers? Eplain our reasoning. 53. OPEN-ENDED MATH Find values of a, b, r, and q such that f() 5 ae r and g() 5 be q are eponential deca functions and f() is an eponential g() growth function. 54. CHALLENGE Eplain wh A 5 P r n 2 nt approimates A 5 Pe rt as n approaches positive infinit. Hint: Let m 5 n r. 2 5 WORKED-OUT SOLUTIONS 496 Chapter 7 Eponential p. WS and Logarithmic Functions 5 STANDARDIZED TEST PRACTICE

22 PROBLEM SOLVING EXAMPLE 4 on p. 494 for Es CAMERA PHONES The number of camera phones shipped globall can be modeled b the function 5.28e.3 where is the number of ears since 997 and is the number of camera phones shipped (in millions). How man camera phones were shipped in 2002? 56. BIOLOGY Scientists used traps to stud the Formosan subterranean termite population in New Orleans. The mean number of termites collected annuall can be modeled b 5 738e 0.345t where t is the number of ears since 989. What was the mean number of termites collected in 999? EXAMPLE 5 on p. 495 for Es FINANCE You deposit $2000 in an account that pas 4% annual interest compounded continuousl. What is the balance after 5 ears? 58. FINANCE You deposit $800 in an account that pas 2.65% annual interest compounded continuousl. What is the balance after 2.5 ears? 59. MULTI-STEP PROBLEM The percent L of surface light that filters down through bodies of water can be modeled b the eponential function L() 5 00e k where k is a measure of the murkiness of the water and is the depth below the surface (in meters). a. A recreational submersible is traveling in clear water with a k-value of about Write and graph an equation giving the percent of surface light that filters down through clear water as a function of depth. b. Use our graph to estimate the percent of surface light available at a depth of 40 meters. c. Use our graph to estimate how deep the submersible can descend in clear water before onl 50% of surface light is available. 60. EXTENDED RESPONSE The growth of the bacteria mcobacterium tuberculosis can be modeled b the function P(t) 5 P 0 e 0.6t where P(t) is the population after t hours and P 0 is the population when t 5 0. a. Model At :00 P.M., there are 30 mcobacterium tuberculosis bacteria in a sample. Write a function for the number of bacteria after :00 P.M. b. Graph Graph the function from part (a). c. Estimate What is the population at 5:00 P.M.? d. Reasoning Describe how to find the population at 3:45 P.M. 7.3 Use Functions Involving e 497

23 6. RATE OF HEALING The area of a wound decreases eponentiall with time. The area A of a wound after t das can be modeled b A 5 A 0 e 20.05t where A 0 is the initial wound area. If the initial wound area is 4 square centimeters, what is the area after 4 das? 62. CHALLENGE The height (in feet) of the Gatewa Arch in St. Louis, Missouri, can be modeled b the function (e /27.7 e 2/27.7 ) where is the horizontal distance (in feet) from the center of the arch. a. Use a graphing calculator to graph the function. How tall is the arch at its highest point? b. About how far apart are the ends of the arch? MIXED REVIEW Solve the equation (p. 5) (p. 5) (p. 292) (p. 292) 67. Ï (p. 452) 68. Ï (p. 452) PREVIEW Prepare for Lesson 7.4 in Es Find the inverse function. (p. 437) 69. f() f() f() f() f() f() QUIZ for Lessons Graph the function. State the domain and range p (p. 478) (p. 486) 3. f() (p. 486) Simplif the epression. (p. 492) 4. 3e 4 p e 3 5. (25e 3 ) 3 6. e 4 5e 7. 8e 5 6e 2 Graph the function. State the domain and range. (p. 492) e e e 2 2. g() 5 4e TV SALES From 997 to 200, the number n (in millions) of black-and-white TVs sold in the United States can be modeled b n (0.85) t where t is the number of ears since 997. Identif the deca factor and the percent decrease. Graph the model and state the domain and range. Estimate the number of black-and-white TVs sold in 999. (p. 478) 3. FINANCE You deposit $200 in an account that pas 4.5% annual interest compounded continuousl. What is the balance after 5 ears? (p. 492) 498 Chapter 7 EXTRA Eponential PRACTICE and Logarithmic for Lesson Functions 7.3, p. 06 ONLINE QUIZ at classzone.com

24 7.4 Evaluate Logarithms and Graph Logarithmic Functions Before You evaluated and graphed eponential functions. Now You will evaluate logarithms and graph logarithmic functions. Wh? So ou can model the wind speed of a tornado, as in Eample 4. Ke Vocabular logarithm of with base b common logarithm natural logarithm You know that and However, for what value of does 2 5 6? Mathematicians define this -value using a logarithm and write 5 log 2 6. The definition of a logarithm can be generalized as follows. KEY CONCEPT For Your Notebook Definition of Logarithm with Base b Let b and be positive numbers with b Þ. The logarithm of with base b is denoted b log b and is defined as follows: log b 5 if and onl if b 5 The epression log b is read as log base b of. This definition tells ou that the equations log b 5 and b 5 are equivalent. The first is in logarithmic form and the second is in eponential form. E XAMPLE Rewrite logarithmic equations Logarithmic Form a. log b. log c. log Eponential Form d. log / Parts (b) and (c) of Eample illustrate two special logarithm values that ou should learn to recognize. Let b be a positive real number such that b Þ. Logarithm of Logarithm of b with Base b log b 5 0 because b 0 5. log b b 5 because b 5 b. GUIDED PRACTICE for Eample Rewrite the equation in eponential form.. log log log log / Evaluate Logarithms and Graph Logarithmic Functions 499

25 E XAMPLE 2 Evaluate logarithms Evaluate the logarithm. a. log 4 64 b. log c. log /5 25 d. log 36 6 Solution To help ou find the value of log b, ask ourself what power of b gives ou. a. 4 to what power gives 64? , so log b. 5 to what power gives 0.2? , so log c. 5 to what power gives 25? , so log / d. 36 to what power gives 6? 36 /2 5 6, so log SPECIAL LOGARITHMS A common logarithm is a logarithm with base 0. It is denoted b log 0 or simpl b log. A natural logarithm is a logarithm with base e. It can be denoted b log e, but is more often denoted b ln. Common Logarithm log 0 5 log Natural Logarithm log e 5 ln Most calculators have kes for evaluating common and natural logarithms. E XAMPLE 3 Evaluate common and natural logarithms Epression Kestrokes Displa Check a. log ø 8 b. ln e ø 0.3 E XAMPLE 4 Evaluate a logarithmic model TORNADOES The wind speed s (in miles per hour) near the center of a tornado can be modeled b s 5 93 log d 65 where d is the distance (in miles) that the tornado travels. In 925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado s center. Solution s 5 93 log d 65 Write function log Substitute 220 for d. ø 93(2.342) 65 Use a calculator. Not drawn to scale Simplif. c The wind speed near the tornado s center was about 283 miles per hour. 500 Chapter 7 Eponential and Logarithmic Functions

26 GUIDED PRACTICE for Eamples 2, 3, and 4 Evaluate the logarithm. Use a calculator if necessar. 5. log log log 2 8. ln WHAT IF? Use the function in Eample 4 to estimate the wind speed near a tornado s center if its path is 50 miles long. INVERSE FUNCTIONS B the definition of a logarithm, it follows that the logarithmic function g() 5 log b is the inverse of the eponential function f() 5 b. This means that: g(f()) 5 log b b 5 and f(g()) 5 b log b 5 E XAMPLE 5 Use inverse properties Simplif the epression. a. 0 log 4 b. log 5 25 Solution a. 0 log b log b 5 b. log log 5 (5 2 ) Epress 25 as a power with base 5. 5 log Power of a power propert 5 2 log b b 5 E XAMPLE 6 Find inverse functions Find the inverse of the function. a. 5 6 b. 5 ln ( 3) REVIEW INVERSES For help with finding inverses of functions, see p Solution a. From the definition of logarithm, the inverse of 5 6 is 5 log 6. b. 5 ln ( 3) Write original function. 5 ln ( 3) Switch and. e 5 3 Write in eponential form. e Solve for. c The inverse of 5 ln ( 3) is 5 e 2 3. GUIDED PRACTICE for Eamples 5 and 6 Simplif the epression log 8. log log 2 64 ln e 4. Find the inverse of Find the inverse of 5 ln ( 2 5). 7.4 Evaluate Logarithms and Graph Logarithmic Functions 50

27 GRAPHING LOGARITHMIC FUNCTIONS You can use the inverse relationship between eponential and logarithmic functions to graph logarithmic functions. KEY CONCEPT For Your Notebook Parent Graphs for Logarithmic Functions The graph of f() 5 log b is shown below for b > and for 0 < b <. Because f() 5 log b and g() 5 b are inverse functions, the graph of f() 5 log b is the reflection of the graph of g() 5 b in the line 5. Graph of f () 5 log b for b > Graph of f () 5 log b for 0 < b < g()5 b (0, ) (, 0) g()5 b (0, ) (, 0) f() 5 log b f()5 log b Note that the -ais is a vertical asmptote of the graph of f() 5 log b. The domain of f() 5 log b is > 0, and the range is all real numbers. E XAMPLE 7 Graph logarithmic functions Graph the function. a. 5 log 3 b. 5 log /2 Solution a. Plot several convenient points, such as (, 0), (3, ), and (9, 2). The -ais is a vertical asmptote. From left to right, draw a curve that starts just to the right of the -ais and moves up through the plotted points, as shown below. b. Plot several convenient points, such as (, 0), (2, 2), (4, 22), and (8, 23). The -ais is a vertical asmptote. From left to right, draw a curve that starts just to the right of the -ais and moves down through the plotted points, as shown below. (3, ) (, 0) 4 (9, 2) (, 0) 3 (2, 2) (4, 22) (8, 23) at classzone.com 502 Chapter 7 Eponential and Logarithmic Functions

28 TRANSLATIONS You can graph a logarithmic function of the form 5 log b ( 2 h) k b translating the graph of the parent function 5 log b. E XAMPLE 8 Translate a logarithmic graph Graph 5 log 2 ( 3). State the domain and range. Solution STEP Sketch the graph of the parent function 5 log 2, which passes through (, 0), (2, ), and (4, 2). STEP 2 Translate the parent graph left 3 units and up unit. The translated graph passes through (22, ), (2, 2), and (, 3). The graph s asmptote is The domain is > 23, and the range is all real numbers. 5 log 2 ( 3) 4 (, 3) (2, 2) (22, ) (4, 2) (2, ) (, 0) 4 5 log 2 GUIDED PRACTICE for Eamples 7 and 8 Graph the function. State the domain and range log log /3 ( 2 3) 8. f() 5 log 4 ( ) EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 3, 33, and 6 5 STANDARDIZED TEST PRACTICE Es. 2, 36, 6, and 62. VOCABULARY Cop and complete: A logarithm with base 0 is called a(n)? logarithm. 2. WRITING Describe the relationship between 5 5 and 5 log 5. EXAMPLE on p. 499 for Es. 3 7 EXPONENTIAL FORM Rewrite the equation in eponential form. 3. log log log log ERROR ANALYSIS Describe and correct the error in rewriting the equation in logarithmic form. 8 log EXAMPLE 2 on p. 500 for Es. 8 9 EVALUATING LOGARITHMS Evaluate the logarithm without using a calculator. 8. log log log log log 9 3. log / log log log / log log log Evaluate Logarithms and Graph Logarithmic Functions 503

29 EXAMPLE 3 on p. 500 for Es EXAMPLE 5 on p. 50 for Es CALCULATING LOGARITHMS Use a calculator to evaluate the logarithm. 20. log 4 2. ln ln log log ln log ln 0 USING INVERSE PROPERTIES Simplif the epression log log log log log log log log MULTIPLE CHOICE Which epression is equivalent to log 00? A B 2 C 0 D 00 EXAMPLE 6 on p. 50 for Es EXAMPLES 7 and 8 on pp for Es FINDING INVERSES Find the inverse of the function log (0.4) log / e ln ( ) log GRAPHING FUNCTIONS Graph the function. State the domain and range log log log / log / log 2 ( 2 3) log f() 5 log 4 ( 2) g() 5 log 6 ( 2 4) h() 5 log 5 ( ) 2 3 CHALLENGE Evaluate the logarithm. (Hint: For each logarithm log b, rewrite b and as powers of the same number.) 54. log log log log 4 28 PROBLEM SOLVING EXAMPLE 4 on p. 500 for Es ALTIMETER 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 in the diagram below. What is the altitude above sea level when the air pressure is 57,000 pascals? 59. CHEMISTRY The ph value for a substance measures how acidic or alkaline the substance is. It is given b the formula ph 5 2log [H ] where H is the hdrogen ion concentration (in moles per liter). Lemon juice has a hdrogen ion concentration of moles per liter. What is its ph value? 5 WORKED-OUT SOLUTIONS 504 Chapter 7 Eponential p. WS and Logarithmic Functions 5 STANDARDIZED TEST PRACTICE

30 60. MULTI-STEP PROBLEM Biologists have found that an alligator s length l (in inches) and weight w (in pounds) are related b the function l ln w Graph the function. Use our graph to estimate the weight of an alligator that is 0 feet long. 6. SHORT RESPONSE The energ magnitude M of an earthquake can be modeled b Peru M (ln E) where E is the amount of energ released (in ergs). a. In 200, a powerful earthquake in Peru, caused b the slippage of two tectonic plates along a fault, released ergs. What was the energ magnitude of the earthquake? b. Find the inverse of the given function. Describe what it represents. Nazca tectonic plate Fault line South American tectonic plate 62. EXTENDED RESPONSE A stud in Florida found that the number of fish species s in a pool or lake can be modeled b the function s (log A) 3.8(log A) 2 where A is the area (in square meters) of the pool or lake. a. Graph Use a graphing calculator to graph the function on the domain 200 A 35,000. b. Estimate Use our graph to estimate the number of fish species in a lake with an area of 30,000 square meters. c. Estimate Use our graph to estimate the area of a lake that contains 6 species of fish. d. Reasoning Describe what happens to the number of fish species as the area of a pool or lake increases. Eplain wh our answer makes sense. 63. CHALLENGE The function s (log d) gives the slope s of a beach in terms of the average diameter d (in millimeters) of sand particles on the beach. Find the inverse of this function. Then use the inverse to estimate the average diameter of the sand particles on a beach with a slope of 0.2. MIXED REVIEW PREVIEW Prepare for Lesson 7.5 in Es Evaluate the epression. (p. 330) p (5 23 ) p 8 3 p (6 22 ) Simplif the epression. Assume all variables are positive. (p. 420) 72. /2 p 2/3 73. (m 9 ) 2/ Î 6 / /2 /2 4 3 Ï (n 4/3 p n 2/5 ) / ( 5 Ï 0 p 3 Ï 9 ) Ï p Ï 7 3 Ï EXTRA PRACTICE for Lesson 7.4, p. 06 Evaluate Logarithms ONLINE and Graph QUIZ Logarithmic at classzone.com Functions 505

31 MIXED REVIEW of Problem Solving STATE TEST PRACTICE classzone.com Lessons MULTI-STEP PROBLEM The graph shown below is a translation of the graph of 5 log 3. (5, 2) (3, ) (, 3) a. Write an equation of the graph. b. Graph the inverse of the function whose graph is shown above. 2. MULTI-STEP PROBLEM When a piece of paper is folded in half, the paper is divided into two regions, each of which has half the area of the paper. If this process is repeated, the number of regions increases while the area of each region decreases. The table below shows the number of regions and the fractional area of each region after each successive fold. Fold number Number of regions 2??? Fractional area of each region 2??? a. Cop and complete the table. b. Write functions giving the number of regions R(n) and the fractional area of each region A(n)after n folds. Tell whether each function represents eponential growth, eponential deca, or neither. 5. EXTENDED RESPONSE A local bank offers certificate of deposit (CD) accounts that ou can use to save mone and earn interest. You are considering two different CDs: a three-ear CD that requires a minimum balance of $500 and pas 2% annual interest, and a five-ear CD that requires a minimum balance of $2000 and pas 3% annual interest. The interest for both accounts is compounded monthl. a. If ou deposit the minimum required amount in each CD, how much mone is in each account at the end of its term? How much interest does each account earn? b. What is the difference in the amounts of interest? c. Describe the benefits and drawbacks of each account. 6. GRIDDED ANSWER Tritium is a radioactive substance used to illuminate eit signs. The amount of tritium disappears over time, a process called radioactive deca. If ou start with a 0 milligram sample of tritium, the number of milligrams left after t ears is given b 5 0e t. How man milligrams of tritium are left after 0 ears? Round our answer to the nearest hundredth. 7. MULTI-STEP PROBLEM The amount of oil collected b a petroleum compan drilling on the U.S. continental shelf can be modeled b ln where is measured in billions of barrels and is the number of wells drilled. 3. OPEN-ENDED The value of one item increases b r% per ear while the value of another item decreases b r% per ear. After 2 ears, both items are worth $00. Choose a value for r and write a function giving each item s value after t ears. Graph both functions in the same coordinate plane. 4. SHORT RESPONSE You deposit $2000 in an account that pas 4% annual interest compounded continuousl. What is our balance after 2 ears? After how man full ears will our balance first eceed $2250? Eplain how ou found our answers. a. Graph the model. b. About how man barrels of oil would ou epect to collect after drilling 000 wells? c. About how man wells need to be drilled to collect 50 billion barrels of oil? 506 Chapter 7 Eponential and Logarithmic Functions

32 7.5 Appl Properties of Logarithms Before You evaluated logarithms. Now You will rewrite logarithmic epressions. Wh? So ou can model the loudness of sounds, as in E. 63. Ke Vocabular base, p. 0 KEY CONCEPT Properties of Logarithms Let b, m, and n be positive numbers such that b Þ. Product Propert log b mn 5 log b m log b n Quotient Propert log m b n 5 log b m 2 log b n Power Propert log b m n 5 n log b m For Your Notebook E XAMPLE Use properties of logarithms AVOID ERRORS Note that in general log m b n Þ log b m log b n and log b mn Þ(log b m)(log b n). Use log 4 3 ø and log 4 7 ø.404 to evaluate the logarithm. a. log log log 7 Quotient propert 4 ø Use the given values of log 4 3 and log Simplif. b. log log 4 (3 p 7) Write 2 as 3 p 7. 5 log 4 3 log 4 7 Product propert ø Use the given values of log 4 3 and log Simplif. c. log log Write 49 as log 4 7 Power propert ø 2(.404) Use the given value of log Simplif. GUIDED PRACTICE for Eample Use log 6 5 ø and log 6 8 ø.6 to evaluate the logarithm.. log log log log Appl Properties of Logarithms 507

33 REWRITING EXPRESSIONS You can use the properties of logarithms to epand and condense logarithmic epressions. REWRITE EXPRESSIONS When ou are epanding or condensing an epression involving logarithms, ou ma assume an variables are positive. E XAMPLE 2 Epand log Epand a logarithmic epression log log log 6 Quotient propert 5 log 6 5 log log 6 Product propert 5 log log 6 2 log 6 Power propert E XAMPLE 3 Standardized Test Practice Which of the following is equivalent to log 9 3 log 2 2 log 3? A log 8 B log 4 C log 8 D log 24 Solution log 9 3 log 2 2 log 3 5 log 9 log log 3 Power propert 5 log (9 p 2 3 ) 2 log 3 Product propert 5 log 9 p 23 3 Quotient propert c The correct answer is D. A B C D 5 log 24 Simplif. GUIDED PRACTICE for Eamples 2 and 3 5. Epand log Condense ln 4 3 ln 3 2 ln 2. CHANGE-OF-BASE FORMULA Logarithms with an base other than 0 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. KEY CONCEPT For Your Notebook Change-of-Base Formula If a, b, and c are positive numbers with b Þ and c Þ, then: log c a 5 log b a log b c In particular, log c a 5 log a log c and log c a 5 ln a ln c. 508 Chapter 7 Eponential and Logarithmic Functions

34 E XAMPLE 4 Use the change-of-base formula Evaluate log 3 8 using common logarithms and natural logarithms. Solution Using common logarithms: log log 8 log 3 ø ø.893 Using natural logarithms: log ln 8 ln 3 ø ø.893 E XAMPLE 5 Use properties of logarithms in real life SOUND INTENSITY 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) 5 0 log I I 0 where I 0 is the intensit of a barel audible sound (about 0 22 watts per square meter). An artist in a recording studio turns up the volume of a track so that the sound s intensit doubles. B how man decibels does the loudness increase? Solution Let I be the original intensit, so that 2I is the doubled intensit. Increase in loudness 5 L(2I) 2 L(I) 5 0 log 2I I log I I 0 Write an epression. Substitute. 5 0 log 2I I 0 2 log I I log 2 log I I 0 2 log I I 0 2 Distributive propert Product propert 5 0 log 2 Simplif. ø 3.0 c The loudness increases b about 3 decibels. Use a calculator. GUIDED PRACTICE for Eamples 4 and 5 Use the change-of-base formula to evaluate the logarithm. 7. log log log log WHAT IF? In Eample 5, suppose the artist turns up the volume so that the sound s intensit triples. B how man decibels does the loudness increase? 7.5 Appl Properties of Logarithms 509

35 7.5 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es., 7, and 7 5 STANDARDIZED TEST PRACTICE Es. 2, 43, 44, 64, 7, and 73. VOCABULARY Cop and complete: To condense the epression log 3 2 log 3, ou need to use the? propert of logarithms. 2. WRITING Describe two was to evaluate log 7 2 using a calculator. EXAMPLE on p. 507 for Es. 3 4 MATCHING EXPRESSIONS Match the epression with the logarithm that has the same value. 3. ln 6 2 ln ln ln 2 6. ln 6 ln 2 A. ln 64 B. ln 3 C. ln 2 D. ln 36 APPROXIMATING EXPRESSIONS Use log 4 ø and log 2 ø.079 to evaluate the logarithm. 7. log 3 8. log log 6 0. log 64. log log 3 3. log 4 4. log 2 EXAMPLE 2 on p. 508 for Es EXPANDING EXPRESSIONS Epand the epression. 5. log ln 5 7. log log log ln log ln log z log ln 2 /3 26. log log 2 Ï 28. ln ln 4 Ï log 3 Ï 9 ERROR ANALYSIS Describe and correct the error in epanding the logarithmic epression. 3. log (log 2 5)(log 2 ) 32. ln ln 8 ln EXAMPLE 3 on p. 508 for Es CONDENSING EXPRESSIONS Condense the epression. 33. log log ln 2 2 ln log log ln 4 ln log 2 4 log log log 4 4 log ln 40 2 ln 2 ln 40. log log ln ln 42. 2(log log 3 4) 0.5 log MULTIPLE CHOICE Which of the following is equivalent to 3 log 4 6? A log 4 8 B log 4 72 C log 4 26 D log Chapter 7 Eponential and Logarithmic Functions

36 44. MULTIPLE CHOICE Which of the following statements is not correct? A log log 3 6 log 3 3 B log log 3 2 log 3 6 C log log 3 4 log 3 3 D log log log 3 3 EXAMPLE 4 on p. 509 for Es CHANGE-OF-BASE FORMULA Use the change-of-base formula to evaluate the logarithm. 45. log log log log log log log log log log log log log log log log ERROR ANALYSIS Describe and correct the error in using the change-of-base formula. log log 3 log 7 EXAMPLE 5 on p. 509 for Es SOUND INTENSITY In Eercises 62 and 63, use the function in Eample Find the decibel level of the sound made b each object shown below. a. b. c. Barking dog: I W/m 2 Ambulance siren: I W/m 2 Bee: I W/m The intensit of the sound of a trumpet is 0 3 watts per square meter. Find the decibel level of a trumpet. 64. OPEN-ENDED MATH For each statement, find positive numbers M, N, and b (with b Þ ) that show the statement is false in general. a. log b (M N) 5 log b M log b N b. log b (M 2 N) 5 log b M 2 log b N CHALLENGE In Eercises 65 68, use the given hint and properties of eponents to prove the propert of logarithms. 65. Product propert log b mn 5 log b m log b n (Hint: Let 5 log b m and let 5 log b n. Then m 5 b and n 5 b.) 66. Quotient propert log b m n 5 log b m 2 log b n (Hint: Let 5 log b m and let 5 log b n. Then m 5 b and n 5 b.) 67. Power propert log b m n 5 n log b m (Hint: Let 5 log b m. Then m 5 b and m n 5 b n.) 68. Change-of-base formula log c a 5 log b a log b c (Hint: Let 5 log b a, 5 log b c, and z 5 log c a. Then a 5 b, c 5 b, and a 5 c z, so that b 5 c z.) 7.5 Appl Properties of Logarithms 5

37 PROBLEM SOLVING EXAMPLE 5 on p. 509 for Es CONVERSATION Three groups of people are having separate conversations in a room. The sound of each conversation has an intensit of watts per square meter. What is the decibel level of the combined conversations in the room? 70. PARKING GARAGE The sound made b each of five cars in a parking garage has an intensit of watts per square meter. What is the decibel level of the sound made b all five cars in the parking garage? 7. SHORT RESPONSE The intensit of the sound TV ads make is ten times as great as the intensit for an average TV show. How man decibels louder is a TV ad? Justif our answer using properties of logarithms. 72. BIOLOGY The loudest animal on Earth is the blue whale. It can produce a sound with an intensit of watts per square meter. The loudest sound a human can make has an intensit of watts per square meter. Compare the decibel levels of the sounds made b a blue whale and a human. 73. EXTENDED RESPONSE The f-stops on a 35 millimeter 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. Then s and f are related b the equation: s 5 log 2 f 2 a. Use Properties Epand the epression for s. b. Calculate The table shows the first eight f-stops on a 35 millimeter camera. Cop and complete the table. Describe the pattern ou observe. f s???????? c. Reasoning Man 35 millimeter cameras have nine f-stops. What do ou think the ninth f-stop is? Eplain our reasoning. 5 WORKED-OUT SOLUTIONS 52 Chapter 7 Eponential p. WS and Logarithmic Functions 5 STANDARDIZED TEST PRACTICE

38 74. CHALLENGE Under certain conditions, the wind speed s (in knots) at an altitude of h meters above a grass plain can be modeled b this function: s(h) 5 2 ln (00h) a. B what factor 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) 5 2 (log h 2). log e MIXED REVIEW Perform the indicated operation. (p. 87) 75. F G F G F G F G 3F G PREVIEW Prepare for Lesson 7.6 in Es Solve the equation. Check for etraneous solutions. (p. 452) 78. Ï Ï Ï 6 5 Ï Ï Ï Ï Ï 9 28 Use a calculator to evaluate the epression. 84. e 8 (p. 492) 85. e 26 (p. 492) 86. e 3.5 (p. 492) 87. e 20.4 (p. 492) 88. log 2 (p. 499) 89. log.8 (p. 499) 90. ln 24 (p. 499) 9. ln 8.49 (p. 499) QUIZ for Lessons Evaluate the logarithm without using a calculator. (p. 499). log log 5 3. log log /2 32 Graph the function. State the domain and range. (p. 499) 5. 5 log ln log 3 ( 4) 2 Epand the epression. (p. 507) 8. log log ln 5 3. log Condense the epression. (p. 507) 2. log log ln 6 ln 4 4. log log ln 2 5 ln Use the change-of-base formula to evaluate the logarithm. (p. 507) 6. log log log log SOUND INTENSITY The sound of an alarm clock has an intensit of I watts per square meter. Use the model L(I) 5 0 log I I 0, where I watts per square meter, to find the alarm clock s loudness L(I). (p. 507) EXTRA PRACTICE for Lesson 7.5, p. 06 ONLINE 7.5 Appl QUIZ Properties at classzone.com of Logarithms 53

39 Use after Lesson Graph Logarithmic Functions classzone.com Kestrokes QUESTION How can ou graph logarithmic functions on a graphing calculator? You can use a graphing calculator to graph logarithmic functions simpl b using the or ke. To graph a logarithmic function having a base other than 0 or e, ou need to use the change-of-base formula to rewrite the function in terms of common or natural logarithms. E XAMPLE Graph logarithmic functions Use a graphing calculator to graph 5 log 2 and 5 log 2 ( 2 3). STEP Rewrite functions Use the change-of-base formula to rewrite each function in terms of common logarithms. 5 log 2 5 log 2 ( 2 3) 5 log log 2 log ( 2 3) 5 log 2 STEP 2 Enter functions Enter each function into a graphing calculator. STEP 3 Graph functions Graph the functions. Y=log(X)/log(2) Y2=(log(X-3)/ log(2))+ Y3= Y4= Y5= Y6= P RACTICE Use a graphing calculator to graph the function.. 5 log log 8 3. f() 5 log log log 2 6. g() 5 log log 3 ( 2) 8. 5 log f() 5 log 4 ( 2 5) log 2 ( 4) log 7 ( 2 5) 3 2. g() 5 log 3 ( 6) REASONING Graph 5 ln. If our calculator did not have a natural logarithm ke, eplain how ou could graph 5 ln using the ke. 54 Chapter 7 Eponential and Logarithmic Functions

40 Solve Eponential and 7.6 Logarithmic Equations Before You studied eponential and logarithmic functions. Now You will solve eponential and logarithmic equations. Wh? So ou can solve problems about astronom, as in Eample 7. Ke Vocabular eponential equation logarithmic equation etraneous solution, p. 52 Eponential equations are equations in which variable epressions occur as eponents. The result below is useful for solving certain eponential equations. KEY CONCEPT Propert of Equalit for Eponential Equations Algebra For Your Notebook If b is a positive number other than, then b 5 b if and onl if 5. Eample If , then 5 5. If 5 5, then E XAMPLE Solve b equating eponents Solve Write original equation. (2 2 ) 5 (2 2 ) 2 3 Rewrite 4 and as powers with base Power of a power propert Solve for. c The solution is. CHECK Propert of equalit for eponential equations Check the solution b substituting it into the original equation Substitute for Simplif Solution checks. GUIDED PRACTICE for Eample Solve the equation Solve Eponential and Logarithmic Equations 55

41 When it is not convenient to write each side of an eponential equation using the same base, ou can solve the equation b taking a logarithm of each side. E XAMPLE 2 Take a logarithm of each side ANOTHER WAY For an alternative method for solving the problem in Eample 2, turn to page 523 for the Problem Solving Workshop. Solve Write original equation. log log 4 5 log 4 5 log log 4 Take log 4 of each side. log b b 5 Change-of-base formula ø.73 Use a calculator. c The solution is about.73. Check this in the original equation. NEWTON S LAW OF COOLING 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 T 5 (T 0 2 T R )e 2rt T R where T R is the surrounding temperature and r is the substance s cooling rate. E XAMPLE 3 Use an eponential model CARS You are driving on a hot da when our car overheats and stops running. It overheats at 2808F and can be driven again at 2308F. If r and it is 808F outside, how long (in minutes) do ou have to wait until ou can continue driving? Solution T 5 (T 0 2 T R )e 2rt T R Newton s law of cooling ( )e t 80 Substitute for T, T 0, T R, and r e t Subtract 80 from each side e t Divide each side b 200. ln ln e t Take natural log of each side ø t ln e 5 log e e 5 60 ø t Divide each side b c You have to wait about 60 minutes until ou can continue driving. GUIDED PRACTICE for Eamples 2 and 3 Solve the equation e Chapter 7 Eponential and Logarithmic Functions

42 SOLVING LOGARITHMIC EQUATIONS Logarithmic equations are equations that involve logarithms of variable epressions. You can use the following propert to solve some tpes of logarithmic equations. KEY CONCEPT For Your Notebook Propert of Equalit for Logarithmic Equations Algebra If b,, and are positive numbers with b Þ, then log b 5 log b if and onl if 5. Eample If log 2 5 log 2 7, then 5 7. If 5 7, then log 2 5 log 2 7. E XAMPLE 4 Solve a logarithmic equation Solve log 5 (4 2 7) 5 log 5 ( 5). log 5 (4 2 7) 5 log 5 ( 5) Write original equation. Propert of equalit for logarithmic equations Subtract from each side. Add 7 to each side. 5 4 Divide each side b 3. c The solution is 4. CHECK Check the solution b substituting it into the original equation. log 5 (4 2 7) 5 log 5 ( 5) Write original equation. log 5 (4 p 4 2 7) 0 log 5 (4 5) Substitute 4 for. log log 5 9 Solution checks. EXPONENTIATING TO SOLVE EQUATIONS The propert of equalit for eponential equations on page 55 implies that if ou are given an equation 5, then ou can eponentiate each side to obtain an equation of the form b 5 b. This technique is useful for solving some logarithmic equations. E XAMPLE 5 Eponentiate each side of an equation Solve log 4 (5 2 ) 5 3. log 4 (5 2 ) 5 3 Write original equation. 4 log 4 (5 2 ) Eponentiate each side using base b log b 5 Add to each side. 5 3 Divide each side b 5. c The solution is 3. CHECK log 4 (5 2 ) 5 log 4 (5 p 3 2 ) 5 log 4 64 Because , log Solve Eponential and Logarithmic Equations 57

43 ELIMINATE CHOICES Instead of solving the equation in Eample 6 directl, ou can substitute each possible answer into the equation to see whether it is a solution. EXTRANEOUS SOLUTIONS 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. E XAMPLE 6 Solution log 2 log ( 2 5) 5 2 log [2( 2 5)] 5 2 Standardized Test Practice What is (are) the solution(s) of log 2 log ( 2 5) 5 2? A 25, 0 B 5 C 0 D 5, 0 Write original equation. Product propert of logarithms 0 log [2( 2 5)] Eponentiate each side using base 0. 2( 2 5) b log b 5 Distributive propert Write in standard form Divide each side b 2. ( 2 0)( 5) 5 0 Factor. 5 0 or 5 25 Zero product propert CHECK Check the apparent solutions 0 and 25 using algebra or a graph. Algebra Substitute 0 and 25 for in the original equation. log 2 log ( 2 5) 5 2 log 2 log ( 2 5) 5 2 log (2 p 0) log (0 2 5) 0 2 log [2(25)] log (25 2 5) 0 2 log 20 log log (20) log (20) 0 2 log Because log (20) is not defined, 25 is not a solution. So, 0 is a solution. Graph Graph 5 log 2 log ( 2 5) and 5 2 in the same coordinate plane. The graphs intersect onl once, when 5 0. So, 0 is the onl solution. c The correct answer is C. A B C D Intersection X=0 Y=2 GUIDED PRACTICE for Eamples 4, 5, and 6 Solve the equation. Check for etraneous solutions. 7. ln (7 2 4) 5 ln (2 ) 8. log 2 ( 2 6) log 5 log ( 2 ) log 4 ( 2) log Chapter 7 Eponential and Logarithmic Functions

44 E XAMPLE 7 Use a logarithmic model ASTRONOMY 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 given b the function M 5 5 log D 2 where D is the diameter (in millimeters) of the telescope s objective lens. If a telescope can reveal stars with a magnitude of 2, what is the diameter of its objective lens? ANOTHER WAY For an alternative method for solving the problem in Eample 7, turn to page 523 for the Problem Solving Workshop. Solution M 5 5 log D 2 Write original equation log D 2 Substitute 2 for M log D Subtract 2 from each side. 25 log D Divide each side b log D Eponentiate each side using base D Simplif. c The diameter is 00 millimeters. at classzone.com GUIDED PRACTICE for Eample 7. WHAT IF? Use the information from Eample 7 to find the diameter of the objective lens of a telescope that can reveal stars with a magnitude of EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5, 35, and 57 5 STANDARDIZED TEST PRACTICE Es. 2, 44, 47, 58, and 60 5 MULTIPLE REPRESENTATIONS E. 59. VOCABULARY Cop and complete: The equation is an eample of a(n)? equation. 2. WRITING When do logarithmic equations have etraneous solutions? EXAMPLE on p. 55 for Es. 3 SOLVING EXPONENTIAL EQUATIONS Solve the equation Solve Eponential and Logarithmic Equations 59

45 EXAMPLE 2 on p. 56 for Es SOLVING EXPONENTIAL EQUATIONS Solve the equation e e e (6) e EXAMPLE 4 on p. 57 for Es SOLVING LOGARITHMIC EQUATIONS Solve the equation. Check for etraneous solutions. 24. log 5 (5 9) 5 log ln (4 2 7) 5 ln ( ) 26. ln ( 9) 5 ln (7 2 8) 27. log 5 (2 2 7) 5 log 5 (3 2 9) 28. log (2 2 ) 5 log (3 3) 29. log 3 (8 7) 5 log 3 (3 38) 30. log 6 (3 2 0) 5 log 6 (4 2 5) 3. log 8 (5 2 2) 5 log 8 (6 2 ) EXAMPLES 5 and 6 on pp for Es EXPONENTIATING TO SOLVE EQUATIONS Solve the equation. Check for etraneous solutions. 32. log ln log log log 2 ( 2 4) log 2 log 2 ( 2 2) log 4 (2) log 4 ( 0) ln ( 3) ln ln (2) log 5 ( 4) log 5 ( ) log 6 3 log 6 ( 2 ) log 3 ( 2 9) log 3 ( 2 3) MULTIPLE CHOICE What is the solution of 3 log 8 (2 7) 8 5 0? A 2.5 B 2.79 C 4 D ERROR ANALYSIS Describe and correct the error in solving the equation log log log 3 6 e log e 5 5 log e e OPEN-ENDED MATH Give an eample of an eponential equation whose onl solution is 4 and an eample of a logarithmic equation whose onl solution is 23. CHALLENGE Solve the equation log 2 ( ) 5 log log 3 5 log p p WORKED-OUT SOLUTIONS 520 Chapter 7 Eponential p. WS and Logarithmic Functions 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

46 PROBLEM SOLVING EXAMPLE 3 on p. 56 for Es COOKING You are cooking beef stew. When ou take the beef stew off the stove, it has a temperature of 2008F. The room temperature is 758F and the cooling rate of the beef stew is r How long (in minutes) will it take to cool the beef stew to a serving temperature of 008F? 55. THERMOMETER As ou are hanging an outdoor thermometer, its reading drops from the indoor temperature of 758F to 378F in one minute. If the cooling rate is r 5.37, what is the outdoor temperature? 56. COMPOUND INTEREST You deposit $00 in an account that pas 6% annual interest. How long will it take for the balance to reach $000 for each given frequenc of compounding? a. Annual b. Quarterl c. Dail 57. RADIOACTIVE DECAY 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 5 00e t. After how man ears will onl 5 grams of radium be present? 58. MULTIPLE CHOICE You deposit $800 in an account that pas 2.25% annual interest compounded continuousl. About how long will it take for the balance to triple? A 24 ears C 48.8 ears B 36 ears D 52.6 ears EXAMPLE 7 on p. 59 for E MULTIPLE REPRESENTATIONS The Richter scale is used for measuring the magnitude of an earthquake. The Richter magnitude R is given b the function R log (0.37E).46 where E is the energ (in kilowatt-hours) released b the earthquake. USA GREECE JAPAN Ocotillo Wells, CA Ma 20, 2005 R = 4. Athens Sept. 7, 999 R = 5.9 Fukuoka March 20, 2005 R = 6.6 a. Making a Graph Graph the function using a graphing calculator. Use our graph to approimate the amount of energ released b each earthquake indicated in the diagram above. b. Solving Equations Write and solve a logarithmic equation to find the amount of energ released b each earthquake in the diagram. 7.6 Solve Eponential and Logarithmic Equations 52

47 60. EXTENDED RESPONSE If X-ras of a fied wavelength strike a material centimeters thick, then the intensit I() of the X-ras transmitted through the material is given b I() 5 I 0 e 2µ, where I 0 is the initial intensit and µ is a number that depends on the tpe of material and the wavelength of the X-ras. The table shows the values of µ for various materials. These µ-values appl to X-ras of medium wavelength. Material Aluminum Copper Lead Value of µ 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() 5 0.3I 0.) b. Repeat part (a) for copper shielding. c. Repeat part (a) for lead shielding. d. Reasoning 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. 6. CHALLENGE You plant a sunflower seedling in our garden. The seedling s height h (in centimeters) after t weeks can be modeled b the function below, which is called a logistic function. h(t) e 20.65t Find the time it takes the sunflower seedling to reach a height of 200 centimeters. Height (cm) h t Weeks MIXED REVIEW PREVIEW Prepare for Lesson 7.7 in Es Solve the sstem using an algebraic method. (p. 60) Determine the possible numbers of positive real zeros, negative real zeros, and imaginar zeros for the function. (p. 379) 65. f() f() f() f() Use finite differences and a sstem of equations to find a polnomial function that fits the data. You ma want to use a graphing calculator to solve the sstem. (p. 393) f () f () Chapter 7 EXTRA Eponential PRACTICE and Logarithmic for Lesson Functions 7.6, p. 06 ONLINE QUIZ at classzone.com

48 LESSON 7.6 Using ALTERNATIVE METHODS Another Wa to Solve Eamples 2 and 7, pp. 56 and 59 MULTIPLE REPRESENTATIONS In Eamples 2 and 7 on pages 56 and 59, respectivel, ou solved eponential and logarithmic equations algebraicall. You can also solve such equations using tables and graphs. P ROBLEM Solve the following eponential equation: 4 5. M ETHOD Using a Table One wa to solve the equation is to make a table of values. STEP Enter the function 5 4 into a graphing calculator. STEP 2 Create a table of values for the function. Y=4^X Y2= Y3= Y4= Y5= Y6= Y7= X Y X=.7 STEP 3 Scroll through the table to find when 5. The table in Step 2 shows that 5 between 5.7 and 5.8. c The solution of 4 5 is between.7 and.8. M ETHOD 2 Using a Graph You can also use a graph to solve the equation. STEP Enter the functions 5 4 and 5 into a graphing calculator. Y=4^X Y2= Y3= Y4= Y5= Y6= Y7= STEP 2 Graph the functions. Use the intersect feature to find the intersection point of the graphs. The graphs intersect at about (.73, ). Use a viewing window of 0 5 and Intersection X= Y= c The solution of 4 5 is about.73. Using Alternative Methods 523

49 P ROBLEM 2 ASTRONOMY 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 given b the function M 5 5 log D 2 where D is the diameter (in millimeters) of the telescope s objective lens. If a telescope can reveal stars with a magnitude of 2, what is the diameter of its objective lens? M ETHOD Using a Table Notice that the problem requires solving the following logarithmic equation: 5 log D One wa to solve this equation is to make a table of values. You can use a graphing calculator to make the table. STEP Enter the function 5 5 log 2 into a graphing calculator. Y=5*log(X)+2 Y2= Y3= Y4= Y5= Y6= Y7= STEP 2 Create a table of values for the function. Make sure that the -values are in the domain of the function ( > 0). X X= Y STEP 3 Scroll through the table of values to find when 5 2. The table shows that 5 2 when X X=00 Y c To reveal stars with a magnitude of 2, a telescope must have an objective lens with a diameter of 00 millimeters. 524 Chapter 7 Eponential and Logarithmic Functions

50 M ETHOD 2 Using a Graph You can also use a graph to solve the equation 5 log D STEP Enter the functions 5 5 log 2 and 5 2 into a graphing calculator. Y=5*log(X)+2 Y2=2 Y3= Y4= Y5= Y6= Y7= STEP 2 Graph the functions. Use the intersect feature to find the intersection point of the graphs. The graphs intersect at (00, 2). Use a viewing window of 0 50 and Intersection X=00 Y=2 c To reveal stars with a magnitude of 2, a telescope must have an objective lens with a diameter of 00 millimeters. P RACTICE EXPONENTIAL EQUATIONS Solve the equation using a table and using a graph e e (3) LOGARITHMIC EQUATIONS Solve the equation using a table and using a graph. 5. log log (23 7) ln log ( 9) ECONOMICS From 998 to 2003, the United States gross national product (in billions of dollars) can be modeled b (.04) where is the number of ears since 998. Use a table and a graph to find the ear when the gross national product was $0 trillion. 0. WRITING In Method of Problem on page 523, eplain how ou could use a table to find the solution of 4 5 more precisel.. WHAT IF? In Problem 2 on page 524, suppose the telescope can reveal stars of magnitude 4. Find the diameter of the telescope s objective lens using a table and using a graph. 2. FINANCE You deposit $5000 in an account that pas 3% annual interest compounded quarterl. How long will it take for the balance to reach $6000? Solve the problem using a table and using a graph. 3. OCEANOGRAPHY The densit d (in grams per cubic centimeter) of seawater with a salinit of 30 parts per thousand is related to the water temperature T (in degrees Celsius) b the following equation: 0.226T d e For deep water in the South Atlantic Ocean off Antarctica, d g/cm 3. Use a table and a graph to find the water s temperature. Using Alternative Methods 525

51 Etension Use after Lesson 7.6 Solve Eponential and Logarithmic Inequalities GOAL Solve eponential and logarithmic inequalities using tables and graphs. In the Problem Solving Workshop on pages , ou learned how to solve eponential and logarithmic equations using tables and graphs. You can use these same methods to solve eponential and logarithmic inequalities. E XAMPLE Solve an eponential inequalit CARS Your famil purchases a new car for $20,000. Its value decreases b 5% each ear. During what interval of time does the car s value eceed $0,000? Solution Let represent the value of the car (in dollars) ears after it is purchased. A function relating and is 5 20,000( 2 0.5), or 5 20,000(0.85). To find the values of for which > 0,000, solve the inequalit 20,000(0.85) > 0,000. METHOD Use a table STEP Enter the function 5 20,000(0.85) into a graphing calculator. Set the starting -value of the table to 0 and the step value to 0.. TABLE SETUP TblStart=0 ntbl=0. Indpnt: Auto Ask Depend: Auto Ask STEP 2 Use the table feature to create a table of values. Scrolling through the table shows that > 0,000 when c The car value eceeds $0,000 for about the first 4.2 ears after it is purchased. To check the solution s reasonableness, note that ø 0,440 when 5 4 and ø 8874 when 5 5. So, 4 < < 5, which agrees with the solution obtained above. METHOD 2 Use a graph Graph 5 20,000(0.85) and 5 0,000 in the same viewing window. Set the viewing window to show 0 8 and 0 25,000. Using the intersect feature, ou can determine that the graphs intersect when ø X X=4.2 Y Intersection X= Y=0000 The graph of 5 20,000(0.85) is above the graph of 5 0,000 when 0 < c The car value eceeds $0,000 for about the first 4.27 ears after it is purchased. 526 Chapter 7 Eponential and Logarithmic Functions

52 E XAMPLE 2 Solve a logarithmic inequalit Solve log 2 2. Solution METHOD Use a table STEP Enter the function 5 log 2 into a graphing calculator as 5 log log 2. Y=log(X)/log(2) Y2= Y3= Y4= Y5= Y6= STEP 2 Use the table feature to create a table of values. Identif the -values for which 2. These -values are given b 0 < 4. Make sure that the -values are reasonable and in the domain of the function ( > 0). c The solution is 0 < 4. X X=4 Y METHOD 2 Use a graph Graph 5 log 2 and 5 2 in the same viewing window. Using the intersect feature, ou can determine that the graphs intersect when 5 4. The graph of 5 log 2 is on or below the graph of 5 2 when 0 < 4. Intersection X=4 Y=2 c The solution is 0 < 4. PRACTICE EXAMPLE on p. 526 for Es. 6 EXAMPLE 2 on p. 527 for Es. 7 2 Solve the eponential inequalit using a table and using a graph > (0.35) (0.96) < (.6) > 235 Solve the logarithmic inequalit using a table and using a graph. 7. log log 5 < 2 9. log log 4 2 > log 2 > log 7 3. FINANCE You deposit $000 in an account that pas 3.5% annual interest compounded monthl. When is our balance at least $200? 4. RATES OF RETURN An investment that earns a rate of return r doubles in value in t ears, where t 5 ln 2 and r is epressed as a decimal. What ln ( r) rates of return will double the value of an investment in less than 0 ears? Etension: Solve Eponential and Logarithmic Inequalities 527

53 7.7 Model Data with an Eponential Function MATERIALS 00 pennies cup graphing calculator Use before Lesson 7.7 classzone.com Kestrokes QUESTION How can ou model data with an eponential function? E XPLORE Collect and record data STEP Make a table Make a table like the one shown to record our results. Number of toss, Number of pennies remaining,???????? STEP 2 Perform an eperiment Record the initial number of pennies in the table, and place the pennies in a cup. Shake the pennies, and then spill them onto a flat surface. Remove all of the pennies showing heads. Count the number of pennies remaining, and record this number in the table. STEP 3 Continue collecting data Repeat Step 2 with the remaining pennies until there are no pennies left to return to the cup. DRAW CONCLUSIONS Use our observations to complete these eercises. What is the initial number of pennies? B what percent would ou epect the number of pennies remaining to decrease after each toss? 2. Use our answers from Eercise to write an eponential function that should model the data in the table. 3. Use a graphing calculator to make a scatter plot of the data pairs (, ). In the same viewing window, graph our function from Eercise 2. Is the function a good model for the data? Eplain. 4. Use the calculator s eponential regression feature to find an eponential function that models the data. Compare this function with the function ou wrote in Eercise Chapter 7 Eponential and Logarithmic Functions

54 7.7 Write and Appl Eponential and Power Functions Before You wrote linear, quadratic, and other polnomial functions. Now You will write eponential and power functions. Wh? So ou can model biolog problems, as in Eample 5. Ke Vocabular power function, In Chapter 2, ou learned that two points determine a line. Similarl, two points determine an eponential curve. p. 428 eponential function, p. 478 E XAMPLE Write an eponential function Write an eponential function 5 ab whose graph passes through (, 2) and (3, 08). Solution STEP Substitute the coordinates of the two given points into 5 ab. 2 5 ab Substitute 2 for and for ab 3 Substitute 08 for and 3 for. STEP 2 Solve for a in the first equation to obtain a 5 2, and substitute this b epression for a in the second equation b 2 b3 Substitute 2 b b 2 Simplif. 9 5 b 2 Divide each side b 2. for a in second equation. 3 5 b Take the positive square root because b > 0. STEP 3 Determine that a 5 2 b So, 5 4 p 3. TRANSFORMING EXPONENTIAL DATA A set of more than two points (, ) fits an eponential pattern if and onl if the set of transformed points (, ln ) fits a linear pattern. Graph of points (, ) Graph of points (, ln ) ln (, 2) ln 5 (ln 2) s2, 2d s22, 4d (0, ) (0, 0) (, 0.69) (2, 20.69) (22, 2.39) The graph is an eponential curve. The graph is a line. 7.7 Write and Appl Eponential and Power Functions 529

55 E XAMPLE 2 Find an eponential model SCOOTERS A store sells motor scooters. The table shows the number of scooters sold during the th ear that the store has been open. Year, Number of scooters sold, Draw a scatter plot of the data pairs (, ln ). Is an eponential model a good fit for the original data pairs (, )? Find an eponential model for the original data. Solution STEP Use a calculator to create a table of data pairs (, ln ) ln STEP 2 Plot the new points as shown. The points lie close to a line, so an eponential model should be a good fit for the original data. ln (7, 4.56) USE POINT-SLOPE FORM Because the aes are and ln, the point-slope form is rewritten as ln 2 5 m( 2 ). The slope of the line through (, 2.48) and (7, 4.56) is: ø STEP 3 Find an eponential model 5 ab b choosing two points on the line, such as (, 2.48) and (7, 4.56). Use these points to write an equation of the line. Then solve for. ln ( 2 ) ln Equation of line Simplif. 5 e Eponentiate each side using base e. 5 e 2.3 (e 0.35 ) 5 8.4(.42) Use properties of eponents. Eponential model (, 2.48) EXPONENTIAL REGRESSION A graphing calculator that performs eponential regression uses all of the original data to find the best-fitting model. E XAMPLE 3 Use eponential regression SCOOTERS Use a graphing calculator to find an eponential model for the data in Eample 2. Predict the number of scooters sold in the eighth ear. Solution Enter the original data into a graphing calculator and perform an eponential regression. The model is (.42). Substituting 5 8 (for ear 8) into the model gives (.42) 8 ø 40 scooters sold. EpReg =a*b^ a= b= Chapter 7 Eponential and Logarithmic Functions

56 GUIDED PRACTICE for Eamples, 2, and 3 Write an eponential function 5 ab whose graph passes through the given points.. (, 6), (3, 24) 2. (2, 8), (3, 32) 3. (3, 8), (6, 64) 4. WHAT IF? In Eamples 2 and 3, how would the eponential models change if the scooter sales were as shown in the table below? Year, Number of scooters sold, WRITING POWER FUNCTIONS Recall from Lesson 6.3 that a power function has the form 5 a b. Because there are onl two constants (a and b), onl two points are needed to determine a power curve through the points. E XAMPLE 4 Write a power function Write a power function 5 a b whose graph passes through (3, 2) and (6, 9). Solution STEP Substitute the coordinates of the two given points into 5 a b. 2 5 a p 3 b Substitute 2 for and 3 for. 9 5 a p 6 b Substitute 9 for and 6 for. STEP 2 Solve for a in the first equation to obtain a b, and substitute this epression for a in the second equation b b Substitute 2 for a in second equation. b p 2 b Simplif b Divide each side b 2. log b Take log 2 of each side. log b Change-of-base formula log ø b Use a calculator. STEP 3 Determine that a ø So, GUIDED PRACTICE for Eample 4 Write a power function 5 a b whose graph passes through the given points. 5. (2, ), (7, 6) 6. (3, 4), (6, 5) 7. (5, 8), (0, 34) 8. REASONING Tr using the method of Eample 4 to find a power function whose graph passes through (3, 5) and (3, 7). What can ou conclude? 7.7 Write and Appl Eponential and Power Functions 53

57 TRANSFORMING POWER DATA A set of more than two points (, ) fits a power pattern if and onl if the set of transformed points (ln, ln ) fits a linear pattern. Graph of points (, ) Graph of points (ln, ln ) 5 /2 (3,.73) (6, 2.45) (4, 2) ln ln 5 ln 2 (.39, 0.69) (, ) (.79, 0.9) (., 0.55) (0, 0) 2 ln The graph is a power curve. The graph is a line. E XAMPLE 5 Find a power model BIOLOGY The table at the right shows the tpical wingspans (in feet) and the tpical weights (in pounds) for several tpes of birds. Draw a scatter plot of the data pairs (ln, ln ). Is a power model a good fit for the original data pairs (, )? Find a power model for the original data. Solution STEP Use a calculator to create a table of data pairs (ln, ln ). ln ln USE POINT-SLOPE FORM The slope of the line is ø STEP 2 STEP 3 Plot the new points as shown. The points lie close to a line, so a power model should be a good fit for the original data. Find a power model 5 a b b choosing two points on the line, such as (.227, 0.525) and (2.28, 2.774). Use these points to write an equation of the line. Then solve for. ln 2 5 m(ln 2 ) ln (ln ) ln ln ln 5 ln e ln e ln 2.5 p e Product Equation when aes are ln and ln Substitute. Simplif. Power propert of logarithms Eponentiate each side using base e. of powers propert Simplif. ln (.227, 0.525) (2.28, 2.774) ln 532 Chapter 7 Eponential and Logarithmic Functions

58 POWER REGRESSION A graphing calculator that performs power regression uses all of the original data to find the best-fitting model. E XAMPLE 6 Use power regression BIOLOGY Use a graphing calculator to find a power model for the data in Eample 5. Estimate the weight of a bird with a wingspan of 4.5 feet. Solution Enter the original data into a graphing calculator and perform a power regression. The model is PwrReg =a*^b a= b= Substituting into the model gives (4.5) 2.87 ø 3.3 pounds. GUIDED PRACTICE for Eamples 5 and 6 9. The table below shows the atomic number and the melting point (in degrees Celsius) for the alkali metals. Find a power model for the data. Alkali metal Lithium Sodium Potassium Rubidium Cesium Atomic number, Melting point, EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es., 23, and 33 5 STANDARDIZED TEST PRACTICE Es. 2, 27, 33, and 35. VOCABULARY Cop and complete: Given a set of more than two data pairs (, ), ou can decide whether a(n)? function fits the data well b making a scatter plot of the points (, ln ). 2. WRITING Eplain how ou can determine whether a power function is a good model for a set of data pairs (, ). EXAMPLE on p. 529 for Es. 3 0 WRITING EXPONENTIAL FUNCTIONS Write an eponential function 5 ab whose graph passes through the given points. 3. (, 3), (2, 2) 4. (2, 24), (3, 44) 5. (3, ), (5, 4) 6. (3, 27), (5, 243) 7. (, 2), (3, 50) 8. (, 40), (3, 640) 9. (2, 0), (4, 0.3) 0. (2, 6.4), (5, 409.6) EXAMPLE 2 on p. 530 for Es. 4 FINDING EXPONENTIAL MODELS Use the points (, ) to draw a scatter plot of the points (, ln ). Then find an eponential model for the data.. (, 8), (2, 36), (3, 72), (4, 44), (5, 288) 2. (, 3.3), (2, 0.), (3, 30.6), (4, 92.7), (5, 280.9) 3. (, 9.8), (2, 2.2), (3, 5.2), (4, 9), (5, 23.8) 4. (,.4), (2, 6.7), (3, 32.9), (4. 6.4), (5, 790.9) 7.7 Write and Appl Eponential and Power Functions 533

59 EXAMPLE 4 on p. 53 for Es WRITING POWER FUNCTIONS Write a power function 5 a b whose graph passes through the given points. 5. (4, 3), (8, 5) 6. (5, 9), (8, 34) 7. (2, 3), (6, 2) 8. (3, 4), (9, 44) 9. (4, 8), (8, 30) 20. (5, 0), (2, 8) 2. (4, 6.2), (7, 23) 22. (3., 5), (6.8, 9.7) EXAMPLE 5 on p. 532 for Es FINDING POWER MODELS Use the given points (, ) to draw a scatter plot of the points (ln, ln ). Then find a power model for the data. 23. (, 0.6), (2, 4.), (3, 2.4), (4, 27), (5, 49.5) 24. (,.5), (2, 4.8), (3, 9.5), (4, 5.4), (5, 22.3) 25. (, 2.5), (2, 3.7), (3, 4.7), (4, 5.5), (5, 6.2) 26. (, 0.8), (2, 0.99), (3,.), (4,.2), (5,.29) 27. MULTIPLE CHOICE Which equation is equivalent to log 5 2? A 5 0(00) B 5 0 C 5 e 2 D 5 e 2 ERROR ANALYSIS Describe and correct the error in writing as a function of. 28. ln e 2 5 e 2 e 5 (e 2 ) e ln 5 3 ln 2 2 ln 5 ln ln e 5 e ln 3 p e 22 5 (3)(0.35) CHALLENGE Take the natural logarithm of both sides of the equations 5 ab and 5 a b. What are the slope and -intercept of the line relating and ln for 5 ab? of the line relating ln and ln for 5 a b? PROBLEM SOLVING GRAPHING CALCULATOR You ma wish to use a graphing calculator to complete the following Problem Solving eercises. EXAMPLES 2, 3, 5, and 6 on pp for Es BIOLOGY Scientists use the circumference of an animal s femur to estimate the animal s weight. The table shows the femur circumference C (in millimeters) and the weight W (in kilograms) for several animals. Animal Giraffe Polar bear Lion Squirrel Otter C (mm) W (kg) a. Draw a scatter plot of the data pairs (ln C, ln W). b. Find a power model for the original data. c. Predict the weight of a cheetah if the circumference of its femur is 68.7 millimeters. 5 WORKED-OUT SOLUTIONS 534 Chapter 7 Eponential p. WS and Logarithmic Functions 5 STANDARDIZED TEST PRACTICE

60 32. ASTRONOMY The table shows the mean distance from the sun (in astronomical units) and the period (in ears) of si planets. Draw a scatter plot of the data pairs (ln, ln ). Find a power model for the original data. Planet Mercur Venus Earth Mars Jupiter Saturn SHORT RESPONSE The table shows the numbers of business and nonbusiness users of instant messaging for the ears Years since Business users (in millions) Non-business users (in millions) a. Find an eponential model for the number of business users over time. b. Eplain how to tell whether a linear, eponential, or power function best models the number of non-business users over time. Then find the bestfitting model. 34. MULTI-STEP PROBLEM The boiling point of water increases with atmospheric pressure. At sea level, where the atmospheric pressure is about 760 millimeters of mercur, water boils at 008C. The table shows the boiling point T of water (in degrees Celsius) for several different values of atmospheric pressure P (in millimeters of mercur). a. Graph Draw a scatter plot of the data pairs (ln P, ln T). b. Model Find a power model for the original data. c. Predict When the atmospheric pressure is 620 millimeters of mercur, at what temperature does water boil? P T EXTENDED RESPONSE Your visual near point is the closest point at which our ees can see an object distinctl. Your near point moves farther awa from ou as ou grow older. The diagram shows the near point (in centimeters) at age (in ears). a. Graph Draw a scatter plot of the data pairs (, ln ). b. Graph Draw a scatter plot of the data pairs (ln, ln ). c. Interpret Based on our scatter plots, does an eponential function or a power function best fit the original data? Eplain our reasoning. d. Model Based on our answer for part (c), write a model for the original data. Use our model to predict the near point for an 80-ear-old person. 7.7 Write and Appl Eponential and Power Functions 535

61 36. CHALLENGE 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. The astronaut s resting pulse rate is 70 beats per minute. Write an eponential model for the data MIXED REVIEW PREVIEW Prepare for Lesson 8. in Es The variables and var directl. Write an equation that relates and. (p. 07) , , , , , , 5 5 Graph the function. State the domain and range. (p. 492) e e 45. f() 5 2e e e g() 5 0.5e 3 Condense the epression. (p. 507) log log log 5 log 4 5. ln 9 ln ln ln 53. log log 5 2 log log log 2 4 log 55. PIZZA A pizza costs $ plus $2 per topping. You have $8. Use an inequalit to find the maimum number of toppings ou can order. (p. 4) QUIZ for Lessons Solve the equation. Check for etraneous solutions. (p. 55) e log 4 (4 7) 5 log 4 6. ln (3 2 2) 5 ln 6 7. log ln log 2 ( 4) 5 5 Write an eponential function 5 ab whose graph passes through the given points. (p. 529) 0. (, 5), (2, 30). (, 4), (2, 32) 2. (2, 5), (3, 45) Write a power function 5 a b whose graph passes through the given points. (p. 529) 3. (4, 8), (9, 23) 4. (3, 2), (0, 36) 5. (5, 4), (, 5) 6. BIOLOGY The average weight (in kilograms) of an Atlantic cod from the Gulf of Maine can be modeled b 5 0.5(.46) where is the age of the cod (in ears). Estimate the age of a cod that weighs 5 kilograms. (p. 55) 536 Chapter 7 EXTRA Eponential PRACTICE and Logarithmic for Lesson Functions 7.7, p. 06 ONLINE QUIZ at classzone.com

62 MIXED REVIEW of Problem Solving Lessons MULTI-STEP PROBLEM The total ependitures in the United States for elementar and secondar schools are shown below. a. Draw a scatter plot of the data pairs (, ln ). b. Draw a scatter plot of the data pairs (ln, ln ). c. Based on our results from parts (a) and (b), find a model for the original data. d. Predict the total ependitures in Years since 997, Total ependitures (billions of dollars), STATE TEST PRACTICE classzone.com 3. OPEN-ENDED Write an eponential function whose graph passes through the point (2, 7). 4. SHORT RESPONSE The total number of miles traveled b motor vehicles in the United States is shown below for various ears. Does an eponential model or a power model best fit the data? Eplain our reasoning. Years since 990, Miles (billions), MULTI-STEP PROBLEM In music, a cent is a unit that is used to epress a small step up or down in pitch. The number c of cents b which two notes differ in pitch is given b c log 2 a b where a and b are the frequencies of the notes a and b. 200 log 2 E4 C4 200 log 2 G4 E4 a. Three notes on the standard scale are C4, E4, and G4. You can compare the difference in the number of cents from C4 to E4 with the difference from E4 to G4 b evaluating this epression: 200 log E4 2 C log G4 2 E4 Write the epression as a single logarithm. b. C4 has a frequenc of 264 hertz, E4 has a frequenc of 330 hertz, and G4 has a frequenc of 396 hertz. Use this information to evaluate our epression from part (a). 5. EXTENDED RESPONSE The effective interest rate is a rate associated with the formula for continuousl compounded interest. The effective interest rate takes into account the effects of compounding on the nominal interest rate (the interest rate in the formula for continuous compounding). The relationship between the effective interest rate E and the nominal interest rate N is given b the equation N 5 ln (E ) where E and N are epressed as decimals. a. What is the effective interest rate for an account that has a nominal interest rate of 5%? Leave our answer in terms of e. b. What is the effective interest rate for an account that has a nominal interest rate of 0%? Leave our answer in terms of e. c. The effective interest rate from part (b) is how man times as great as the effective interest rate from part (a)? Write our answer as a ratio in terms of e. d. Show that the ratio from part (c) is equal to e GRIDDED ANSWER You invest $4000 in an account that pas 2% annual interest compounded continuousl. To the nearest ear, how long will it take to earn $000 interest? Mied Review of Problem Solving 537

63 7 CHAPTER SUMMARY Algebra classzone.com Electronic Function Librar Big Idea BIG IDEAS Graphing Eponential and Logarithmic Functions For Your Notebook Parent functions for eponential functions have the form 5 b. Parent functions for logarithmic functions have the form 5 log b. Eponential Growth Eponential Deca Logarithmic Functions (0, ) 5 b 5 b (0, ) 5 log b (, 0) 5 log b (, 0) b > 0 < b < b > 0 < b < Big Idea 2 Solving Eponential and Logarithmic Equations Solving an Eponential Equation If each side can be written using the same base, equate eponents (3 2 ) If each side cannot be written using the same base, take a logarithm of each side log log 6 5 log 5 5 log 6 ø.5 Solving a Logarithmic Equation If the equation has the form log b 5 log b, use the fact that 5. log 2 (4 2 2) 5 log If a logarithm is set equal to a constant, eponentiate each side. log 5 ( ) Big Idea 3 Writing and Appling Eponential and Power Functions Write an Eponential Model An eponential model fits a set of data pairs (, ) if a linear model fits the set of data pairs (, ln ). Write a Power Model A power model fits a set of data pairs (, ) if a linear model fits the set of data pairs (ln, ln ). 5 b ln 5 b ln ln 5 b ln ln 5 ln b ln 538 Chapter 7 Eponential and Logarithmic Functions

64 7 CHAPTER REVIEW REVIEW KEY VOCABULARY classzone.com Multi-Language Glossar Vocabular practice eponential function, p. 478 eponential growth function, p. 478 growth factor, p. 478 asmptote, p. 478 eponential deca function, p. 486 deca factor, p. 486 natural base e, p. 492 logarithm of with base b, p. 499 common logarithm, p. 500 natural logarithm, p. 500 eponential equation, p. 55 logarithmic equation, p. 57 VOCABULARY EXERCISES. What is the asmptote of the graph of the function ? 2. Identif the deca factor in the model 5 7.2(0.89). 3. WRITING Eplain the meaning of log b. 4. Cop and complete: A logarithm with base e is called a(n)? logarithm. 5. Is 5 (.4) an eponential function or a power function? Eplain. REVIEW EXAMPLES AND EXERCISES Use the review eamples and eercises below to check our understanding of the concepts ou have learned in each lesson of Chapter Graph Eponential Growth Functions pp E XAMPLE Graph 5 2 p State the domain and range. Begin b sketching the graph of 5 2 p 3, which passes through (0, 2) and (, 6). Then translate the graph right 2 units and up 3 units. Notice that the translated graph passes through (2, 5) and (3, 9). The graph s asmptote is the line 5 3. The domain is all real numbers, and the range is > ? (0, 2) 5 2? 3 (, 6) (2, 5) (3, 9) EXAMPLES, 2, 3, and 5 on pp for Es. 6 9 EXERCISES Graph the function. State the domain and range (2.5) 8. f() 5 23 p FINANCE You deposit $500 in an account that pas 7% annual interest compounded dail. Find the balance after 2 ears. Chapter Review 539

65 7 7.2 CHAPTER REVIEW Graph Eponential Deca Functions pp E XAMPLE Graph State the domain and range. Begin b sketching the graph of , which passes through (0, 2) and, 2 2. Then translate the graph left 2 units and down 2 units. Notice that the translated graph passes through (22, 0) and 2, The graph s asmptote is the line The domain is all real numbers, and the range is > 22. (22, 0) 3 s2, 2 2d 25 4 (0, 2) s, 2d 5 2 c c EXAMPLES, 2, and 3 on pp for Es. 0 2 EXERCISES Graph the function. State the domain and range f() 5 2(0.8) Use Functions Involving e pp E XAMPLE Graph 5 e 0.25( 2 ) 2 5. State the domain and range. Because a 5 is positive and r is positive, the function is an eponential growth function. Begin b sketching the graph of 5 e Translate the graph right unit and down 5 units. The domain is all real numbers, and the range is > e e 0.25(2) 2 5 (3, 2.2) (0, ) 2 (, 24) (4, 22.88) EXAMPLES 3 and 5 on pp for Es. 3 6 EXERCISES Graph the function. State the domain and range e e f() 5 e 20.4( 2) 6 6. PHYSIOLOGY Nitrogen-3 is a radioactive isotope of nitrogen used in a phsiological test called positron emission tomograph (PET). A tpical PET scan begins with 6.9 picograms of nitrogen-3 ( picogram grams). The number N of picograms of nitrogen-3 remaining after t minutes can be modeled b N 5 6.9e t. How man picograms of nitrogen-3 remain after 0 minutes? 540 Chapter 7 Eponential and Logarithmic Functions

66 classzone.com Chapter Review Practice 7.4 Find Logarithms and Graph Logarithmic Functions pp E XAMPLE Evaluate the logarithm. a. log b. log 0.00 c. log 25 5 d. log 2 64 To help ou find the value of log b, ask ourself what power of b gives ou. a. 5 to what power gives 625? b. 0 to what power gives 0.00? , so log , so log c. 25 to what power gives 5? d. 2 to what power gives 64? 25 /3 5 5, so log , so log EXAMPLES 2, 4, 7, and 8 on pp for Es EXERCISES Evaluate the logarithm without using a calculator. 7. log log 7 9. log / log 25 5 Graph the function. State the domain and range log / log f() 5 ln ( 2 ) BIOLOGY Researchers have found that after 25 ears of age, the average size of the pupil in a person s ee decreases. The relationship between pupil diameter d (in millimeters) and age a (in ears) can be modeled b d ln a What is the average diameter of a pupil for a person 25 ears old? 50 ears old? 7.5 Appl Properties of Logarithms pp E XAMPLES Epand the epression. Condense the epression. log log log log log log log log 5 6 log 5 2 log log log 5 6 log log 5 5 log 3 32 EXAMPLES 2 and 3 on p. 508 for Es EXERCISES Epand the epression. 25. log ln log 8 4 Condense the epression. 28. ln log 7 4 log ln ln 3. 2 ln 3 5 ln 2 2 ln 8 Chapter Review 54

67 7 7.6 CHAPTER REVIEW Solve Eponential and Logarithmic Equations pp E XAMPLE Solve the equation. a b. log 2 (3 2 7) 5 5 log log log 2 (3 2 7) log log 2 5 ø log 7 EXAMPLES 2, 5, and 6 on pp for Es EXERCISES Solve the equation. Check for etraneous solutions log 3 (2 2 5) ln ln ( 2) Write and Appl Eponential and Power Functions pp E XAMPLE Write an eponential function 5 ab whose graph passes through (2, 2) and (3, 32). Substitute the coordinates of the two given points into 5 ab. 2 5 ab 2 Substitute 2 for and 2 for ab 3 Substitute 32 for and 3 for. Solve for a in the first equation to obtain a 5 2b, and substitute this epression for a in the second equation (2b)b 3 Substitute 2b for a in second equation b 4 Product of powers propert 6 5 b 4 Divide each side b b Take the positive fourth root because b > 0. Because b 5 2, it follows that a 5 2(2) 5 4. So, 5 4 p 2. EXAMPLES and 5 on pp for Es EXERCISES Write an eponential function 5 ab whose graph passes through the points. 35. (3, 8), (5, 2) 36. (22, 2), (, 0.25) 37. (2, 9), (4, 324) 38. SPORTING GOODS A store begins selling a new tpe of basketball shoe. The table shows sales of the shoe over time. Find a power model for the data. Week, Pairs sold, Chapter 7 Eponential and Logarithmic Functions

68 7 Graph CHAPTER TEST the function. State the domain and range p f() 5 25 p (0.25) g() e e h() 5 3 e Evaluate the logarithm without using a calculator. 0. log log log 6 Graph the function. State the domain and range log ln f() 5 log ( 3) 2 Condense the epression ln ln 4 7. log log log 5 log 2 2 log 3 Use the change-of-base formula to evaluate the logarithm. 9. log log log 9 45 Solve the equation. Check for etraneous solutions log ( 2 4) log 4 log 4 ( 6) Write an eponential function 5 ab whose graph passes through (2, 48) and (2, 6). 26. Write a power function 5 a b whose graph passes through (3, 8) and (6, 5). 27. LANDSCAPING From 996 to 200, the number of households that purchased lawn and garden products at home gardening centers increased b about 4.85% per ear. In 996, about 62 million households purchased lawn and garden products. Write a function giving the number of households H (in millions) that purchased lawn and garden products t ears after FINANCE You deposit $2500 in an account that pas 3.5% annual interest compounded continuousl. What is the balance after 8 ears? 29. EARTH SCIENCE Rivers and streams carr small particles of sediment downstream. The table shows the diameter (in millimeters) of several particles of sediment and the speed (in meters per second) of the current needed to carr each particle downstream. a. Draw a scatter plot of the data pairs (ln, ln ). b. Find a power model for the original data. Estimate the speed of the current needed to carr a particle with a diameter of 20 millimeters downstream. Tpe of sediment Mud Gravel Coarse gravel 0.75 Pebbles Small stones Chapter Test 543

69 7 Standardized TEST PREPARATION MULTIPLE CHOICE QUESTIONS If ou have difficult solving a multiple choice problem directl, ou ma be able to use another approach to eliminate incorrect answer choices and obtain the correct answer. P ROBLEM Which eponential function has a graph that passes through the points (2, 22) and (4, 248)? A 5 3 p 2 B 5 2 Ï 3 p 2 C p 3 D 5 23 p 2 METHOD SOLVE DIRECTLY Substitute the coordinates of the two points into 5 ab and solve the resulting sstem. STEP Substitute the coordinates of the points into 5 ab ab 2 Substitute (2, 22) ab 4 Substitute (4, 248). STEP 2 Solve the first equation for a. 22 b 2 5 a Divide each side b b 2. STEP 3 Substitute 22 for a in the second b 2 equation and solve for b b 2 2 b4 Substitute b 2 Simplif. 4 5 b 2 Divide each side b b Take the positive square root because b > 0. STEP 4 Substitute the value of b into a 5 22 to b 2 find the value of a. a b The equation is 5 23 p 2. c The correct answer is D. A B C D METHOD 2 ELIMINATE CHOICES Another method is to check whether both of the points are solutions of the equations given in the answer choices. Substitute the coordinates of the points into the equation in each answer choice. You can stop as soon as ou realize that one of the points is not a solution. Choice A: Choice B: 5 3 p p Þ Ï 3 p Ï 3 p Þ 24 Ï 3 Choice C: p p p p Þ 208 Choice D: 5 23 p p p p c The correct answer is D. A B C D 544 Chapter 7 Eponential and Logarithmic Functions

70 P ROBLEM 2 You bu a new personal computer for $600. It is estimated that the computer s value will decrease b 50% each ear. After about how man ears will the computer be worth $250? A 3 ears B 2 ears C 2 2 ears D 3 ears 3 METHOD SOLVE DIRECTLY Write and solve an equation to find the time it takes for the computer to depreciate to $250. Let be the value (in dollars) of the computer t ears after the purchase. An eponential deca model for the value is: 5 a( 2 r) t ( 2 0.5) t 0.56 ø (0.5) t log log 0.5 (0.5) t log 0.56 log 0.5 ø 2.68 ø t The computer will be worth $250 after about 2.68 ø ears. c The correct answer is C. A B C D METHOD 2 ELIMINATE CHOICES Use estimation to find how long it will take for the computer to depreciate to $250. The computer depreciates b 50% each ear. After 0 ears, it is worth $600. After ear, it will be worth 0.5($600) 5 $800. After 2 ears, it will be worth 0.5($800) 5 $400. After 3 ears, it will be worth 0.5($400) 5 $200. The computer will be worth $250 at some time strictl between 2 and 3 ears after purchase. So, ou can eliminate choices A, B, and D. c The correct answer is C. A B C D PRACTICE Eplain wh ou can eliminate the highlighted answer choice.. Which power function has a graph that passes through the points (22, 26) and (, 2)? A B C 5 2 /2 D For which equation is 4 a solution? A e B log 2 ( 2) 5 log 2 2 C D ln ( 2) ln 5 3. What is the domain of the function 5 25 p 2 2? A All real numbers B All real numbers ecept 22 C All real numbers less than 0 D All real numbers greater than 25 Standardized Test Preparation 545

71 7 Standardized TEST PRACTICE MULTIPLE CHOICE. In 999, the tuition for one ear at Harvard Universit was $22,054. During the net 4 ears, the tuition increased b 4.25% each ear. Which model represents the situation? A 5 22,054(0.0425) t B 5 22,054(0.425) t C 5 22,054(.0425) t D 5 22,054(.425) t 2. You deposit $000 in an account that pas 3% annual interest. How much more interest is earned after 2 ears if the interest is compounded dail than if the interest is compounded monthl? A $.07 B $.27 C $6.76 D $ The graph of which function is shown? (0, 2) 6. The population of the United States is epected to increase b 0.9% each ear from 2003 to 204. The U.S. population was about 290 million in To the nearest million, what is the projected population for 200? A 295 million B 309 million C 574 million D 489 million 7. Which epression is equivalent to Ï 00e 6? A 0e 3 B 7e 3e 2 C (2e ) 3 D 0e Ï 6 8. Which function is an eponential deca function? A 5 2 p 5 B 5 22 p 5 C 5 2e D 5 2(0.5) 9. Which function is the inverse of 5 e 2 2 3? A 5 2 ln ( 3) B 5 2 ln ( 2 3) (, 27) A 5 2 p 4 B 5 22 p 4 C 5 2 p 4 2 D 5 22 p Which function can be obtained b translating the graph of 5 log 3 ( 2) 2 4 left unit? A 5 log 3 ( 2) 2 5 B 5 log 3 ( 2) 2 3 C 5 log 3 ( ) 2 4 D 5 log 3 ( 3) Which epression is equivalent to 3 log log 3? A log 9 B 4 log 3 C log 3 3 D log ( 3 3) C 5 ln ( 3) 2 D 5 ln ( 2 3) 2 0. What is (are) the solution(s) of the equation log ( 3) log 5? A 2, 25 B 2 C 5 D 2, 5. The graph of which function is shown? (, 4) (4, 8) A B C D 5 2 /3 546 Chapter 7 Eponential and Logarithmic Functions

72 STATE TEST PRACTICE classzone.com GRIDDED ANSWER 2. You deposit $500 in an account that pas 3.5% annual interest compounded continuousl. To the nearest dollar, how much interest have ou earned after ear? 3. The model 5 7.7e 0.4 gives the number (in thousands per cubic centimeter) of bacteria in a liquid culture after hours. After how man hours will there be 50,000 bacteria per cubic centimeter? Round our answer to the nearest tenth of an hour. 4. What is the solution of the equation ? 5. What is the value of log log 2 3? 6. The graph of an eponential function f() 5 ab passes through the points (0, 8) and (2, 2). What is the value of f(5)? 7. What is the solution of the equation log 3 (5 3) 5 5? SHORT RESPONSE 8. To make fudge, ou must heat the miture to 2328F and then let it cool at room temperature, 688F. According to the recipe, it should take 20 minutes for the fudge to reach 08F. To the nearest thousandth, what is the cooling rate of the fudge? If ou are in a hurr, how could ou increase the cooling rate? Eplain our reasoning. 9. A movie grosses $37 million in its first week of release. The weekl gross decreases b 30% each week. Write an eponential deca model for the weekl gross in week. What is a reasonable domain for this situation? Eplain our reasoning. 20. A double eponential function is a function of the form 5 a a. Graph the functions 5 2 and in the same coordinate plane. Which function grows more quickl? Eplain. EXTENDED RESPONSE 2. The table shows the number of transistors per integrated circuit for computers introduced in various ears. a. Let be the number of ears since 974, and let be the number of transistors. Draw a scatter plot of the data pairs (, ln ). b. Find an eponential model for the original data. c. Use our model to predict the number of transistors per integrated circuit in d. In 965, Gordon Moore made an observation that became known as Moore s law. Moore s law states that the number of transistors per integrated circuit would double about ever 8 months. According to our model, does Moore s law hold? Eplain our reasoning. 22. For a temperature of 608F and a height of h feet above sea level, the air pressure P (in pounds per square inch) can be modeled b the equation P 5 4.7e h, and the air densit D (in pounds per cubic foot) can be modeled b the equation D e h. a. What is the air densit at 0,000 feet above sea level? b. To the nearest foot, at what height is the air pressure 2 pounds per square inch? c. What is the relationship between air pressure and air densit at 608F? Eplain our reasoning. d. One rule of thumb states that air pressure decreases b about % for ever 80 meter increase in altitude. Do ou agree with this? Eplain. Year Transistors 974 6, , , , ,200, ,00, ,500, ,500, ,000, ,000,000 Standardized Test Practice 547