Exponential and Logarithmic Functions

Transcription

1 7 Eponential and Logarithmic Functions In this chapter ou will stud two tpes of nonalgebraic functions eponential functions and logarithmic functions. Eponential and logarithmic functions are widel used in describing economic and phsical phenomena such as compound interest, population growth, memor retention, and deca of radioactive material. In this chapter, ou should learn the following. How to recognize, evaluate, and graph eponential functions. (7.) How to recognize, evaluate, and graph logarithmic functions. (7.) How to use properties of logarithms to evaluate, rewrite, epand, or condense logarithmic epressions. (7.) How to solve eponential and logarithmic equations. (7.) How to use eponential growth models, eponential deca models, Gaussian models, logistic growth models, and logarithmic models to solve real-life problems. (7.5) Juniors Bildarchiv / Alam Given data about four-legged animals, how can ou find a logarithmic function that can be used to relate an animal s weight and its lowest galloping speed? (See Section 7., Eercise 96.) = e = e = + e = + ln You can use eponential and logarithmic functions to model man real-life situations. You will learn about the tpes of data that are best represented b the different models. (See Section 7.5.) 7 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

2 7 Chapter 7 Eponential and Logarithmic Functions 7. Eponential Functions and Their Graphs Recognize and evaluate eponential functions with base a. Graph eponential functions. Recognize, evaluate, and graph eponential functions with base e. Use eponential functions to model and solve real-life problems. Eponential Functions So far, this tet has dealt mainl with algebraic functions, which include polnomial functions and rational functions. In this chapter, ou will stud two tpes of nonalgebraic functions eponential functions and logarithmic functions. These functions are eamples of transcendental functions. DEFINITION OF EXPONENTIAL FUNCTION The eponential function f with base a is denoted b f a where a > 0, a, and is an real number. The base a is ecluded because it ields f. This is a constant function, not an eponential function. You have evaluated a for integer and rational values of. For eample, ou know that 6 and. However, to evaluate for an real number, ou need to interpret forms with irrational eponents. For the purposes of this tet, it is sufficient to think of a (where.56) as the number that has the successivel closer approimations a., a., a., a., a.,.... Graphs of Eponential Functions The graphs of all eponential functions have similar characteristics, as shown in Eamples through. g() = f() = Figure 7. EXAMPLE Graphs of a In the same coordinate plane, sketch the graphs of f and g. Solution The table below lists some values for each function, and Figure 7. shows the graphs of the two functions. Note that both graphs are increasing. Moreover, the graph of g is increasing more rapidl than the graph of f The table feature of a graphing utilit could be used to epand the table. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

3 7. Eponential Functions and Their Graphs 7 G() = F() = Figure 7. EXAMPLE Graphs of a In the same coordinate plane, sketch the graphs of F and G. Solution The table below lists some values for each function, and Figure 7. shows the graphs of the two functions. Note that both graphs are decreasing. Moreover, the graph of G is decreasing more rapidl than the graph of F. 0 6 In Eample, note that b using the properties of eponents, the functions F and G can be rewritten with positive eponents. F 6 and 8 6 G STUDY TIP Notice that the range of an eponential function is 0,, which means that a > 0 for all values of. EXPLORATION Use a graphing utilit to graph a for a, 5, and 7 in the same viewing window. (Use a viewing window in which and 0.) How do the graphs compare with each other? Which graph is on the top in the interval, 0? Which is on the bottom? Which graph is on the top in the interval 0,? Which is on the bottom? Repeat this eperiment with the graphs of b for b, 5, and 7. (Use a viewing window in which and 0.) What can ou conclude about the shape of the graph of b and the value of b? Comparing the functions in Eamples and, observe that F f and G g. Consequentl, the graph of F is a reflection (in the -ais) of the graph of f. The graphs of G and g have the same relationship. The graphs in Figures 7. and 7. are tpical of the eponential functions a and a. The have one -intercept and one horizontal asmptote (the -ais), and the are continuous. The basic characteristics of these eponential functions are summarized in Figures 7. and 7.. Figure 7. (0, ) Figure 7. = a = a (0, ) Graph of a, a > Domain:, Range: 0, -intercept: 0, Increasing -ais is a horizontal asmptote a 0 as. Continuous Graph of a, a > Domain:, Range: 0, -intercept: 0, Decreasing -ais is a horizontal asmptote a 0 as. Continuous Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

4 7 Chapter 7 Eponential and Logarithmic Functions In the following eample, notice how the graph of a can be used to sketch the graphs of functions of the form f b ± a c. EXAMPLE Transformations of Graphs of Eponential Functions Use the graph of f to describe the transformation that ields the graph of g. a. g b. g c. g d. g Solution a. Because g f, the graph of g can be obtained b shifting the graph of f one unit to the left. See Figure 7.5(a). b. Because g f, the graph of g can be obtained b shifting the graph of f down two units. See Figure 7.5(b). c. Because g f, the graph of g can be obtained b reflecting the graph of f in the -ais. See Figure 7.5(c). d. Because g f, the graph of g can be obtained b reflecting the graph of f in the -ais. See Figure 7.5(d). g() = + f() = f() = g() = (a) Horizontal shift to the left (b) Vertical shift downward f() = g() = f() = g() = (c) Reflection in the -ais (d) Reflection in the -ais Figure 7.5 In Figure 7.5, notice that the transformations in parts (a), (c), and (d) keep the -ais as a horizontal asmptote, but the transformation in part (b) ields a new horizontal asmptote of. Also, be sure to note how the -intercept is affected b each transformation. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

5 7. Eponential Functions and Their Graphs 75 (, e) The Natural Base e In man applications, the most convenient choice for a base is the irrational number e This number is called the natural base. The function given b f e is called the natural eponential function. Its graph is shown in Figure 7.6. Be sure ou see that for the eponential function given b f e, e is the constant , whereas is the variable. f() = e (, e ) (0, ) (, e ) Figure 7.6 THEOREM 7. A LIMIT INVOLVING e The following limits eist and are equal. The real number that is the limit is defined to be e lim e lim 0 e THE NUMBER e The smbol e was first used b mathematician Leonhard Euler to represent the base of natural logarithms in a letter to another mathematician, Christian Goldbach, in 7. STUDY TIP The choice of e as a base for eponential functions ma seem anthing but natural. In Section 8., ou will see more clearl wh e is the convenient choice for a base. EXAMPLE Graphing Natural Eponential Functions Sketch the graph of each natural eponential function. a. f e 0. b. g e 0.58 Solution To sketch these two graphs, ou can use a graphing utilit to construct a table of values, as shown below. After constructing the table, plot the points and connect them with smooth curves, as shown in Figure 7.7. Note that the graph in part (a) is increasing whereas the graph in part (b) is decreasing. 0 f g f() = e g() = e 0.58 (a) Figure 7.7 (b) Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

6 76 Chapter 7 Eponential and Logarithmic Functions Applications One of the most familiar eamples of eponential growth is that of an investment earning continuousl compounded interest. Using eponential functions, ou can develop a formula for the balance in an account that pas compound interest, and show how it leads to continuous compounding. Suppose a principal P is invested at an annual interest rate r, compounded once a ear. If the interest is added to the principal at the end of the ear, the new balance is P P Pr P r. This pattern of multipling the previous principal b r is then repeated each successive ear, as shown below. Year 0.. t Balance After Each Compounding P P P P r P P r P r r P r P P r P r r P r.. P t P r t To accommodate more frequent (quarterl, monthl, or dail) compounding of interest, let n be the number of compoundings per ear and let t be the number of ears. Then the rate per compounding is r n and the account balance after t ears is P A P r n nt. Amount (balance) with n compoundings per ear If ou let the number of compoundings n increase without bound, the process approaches what is called continuous compounding. In the formula for n compoundings per ear, let m n r. This produces A P r n nt P r mr mrt P m mrt Amount with n compoundings per ear Substitute mr for n. Simplif. P m m rt. Propert of eponents As m increases without bound, m m approaches e. From this, ou can conclude that the formula for continuous compounding is A Pe rt. Substitute e for m m. STUDY TIP Be sure ou see that the annual interest rate must be epressed in decimal form when using the compound interest formula. For instance, 6% should be epressed as FORMULAS FOR COMPOUND INTEREST After t ears, the balance A in an account with principal P and annual interest rate r (in decimal form) is given b the following formulas.. For n compoundings per ear: A P r n nt. For continuous compounding: A Pe rt Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

7 7. Eponential Functions and Their Graphs 77 NOTE In Eample 5, note that continuous compounding ields more than quarterl, monthl, or dail compounding. This is tpical of the two tpes of compounding. That is, for a given principal, interest rate, and time, continuous compounding will alwas ield a larger balance than compounding n times a ear. EXPLORATION Use a graphing utilit to make a table of values that shows the amount of time it would take to double the investment in Eample 5 using continuous compounding. EXAMPLE 5 Compound Interest A total of $,000 is invested at an annual interest rate of 9%. Find the balance after 5 ears if it is compounded a. quarterl. b. monthl. c. dail. d. continuousl. Solution a. For quarterl compoundings, ou have n. So, in 5 ears at 9%, the balance is A P r n nt b. For monthl compoundings, ou have n. So, in 5 ears at 9%, the balance is A P r n nt, (5), (5) $8,76.. $8, c. For dail compoundings, ou have n 65. So, in 5 ears at 9%, the balance is, $8, A P n r nt d. For continuous compounding, the balance is A Pe rt,000e 0.09(5) $8, EXAMPLE 6 Radioactive Deca The half-life of radioactive radium 6 Ra is about 599 ears. That is, for a given amount of radium, half of the original amount will remain after 599 ears. After another 599 ears, one-quarter of the original amount will remain, and so on. Let represent the mass, in grams, of a quantit of radium. The quantit present after t ears, then, is 5 t 599. a. What is the initial mass (when t 0)? b. How much of the initial mass is present after 500 ears? Algebraic Solution Graphical Solution a. 5 Write original equation. t 599 Use a graphing utilit to graph 5 t 599. a. Use the value feature or the zoom and trace features of the graphing 5 Substitute 0 for t utilit to determine that when 0, the value of is 5, as shown in Figure 7.8(a). So, the initial mass is 5 grams. 5 Simplif. b. Use the value feature or the zoom and trace features of the graphing utilit to determine that when 500, the value of is about 8.6, So, the initial mass is 5 grams. as shown in Figure 7.8(b). So, about 8.6 grams is present after b. 5 Write original equation. t ears Substitute 500 for t. 5 Simplif Use a calculator. So, about 8.6 grams is present after 500 ears. 0 0 (a) Figure (b) 5000 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

8 78 Chapter 7 Eponential and Logarithmic Functions 7. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 6, fill in the blanks.. Polnomial and rational functions are eamples of functions.. Eponential and logarithmic functions are eamples of nonalgebraic functions, also called functions.. You can use of the graph of a to sketch the graphs of functions of the form f b ± a c.. The eponential function given b f e is called the function, and the base e is called the base. 5. To find the amount A in an account after t ears with principal P and an annual interest rate r compounded n times per ear, ou can use the formula. 6. To find the amount A in an account after t ears with principal P and an annual interest rate r compounded continuousl, ou can use the formula. In Eercises 7 0, evaluate the function at the indicated value of. Round our result to three decimal places. Function Value f 0.9 f. f 5. In Eercises, match the eponential function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) (c) f (0, ) (0, ) (d). f. f. f. f 6 (0, ) 6 (0, ) 6 In Eercises 5 0, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. 5. f 6. f 7. f 6 8. f 6 9. f 0. f In Eercises 6, use the graph of f to describe the transformation that ields the graph of g f, f, f, f 0, f 7, f 0., g g g g 0 g 7 g 0. 5 In Eercises 7 0, use a graphing utilit to graph the eponential function In Eercises, evaluate the function at the indicated value of. Round our result to three decimal places..... Function Value h e f e. f e 5 0 f.5e 0 In Eercises 5 0, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. 5. f e 6. f e 7. f e 8. f e f e 0. f e 5 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

9 7. Eponential Functions and Their Graphs 79 In Eercises 6, use a graphing utilit to graph the eponential function s t e 0.t. s t e 0.t 5. g e 6. h e WRITING ABOUT CONCEPTS In Eercises 7 50, use properties of eponents to determine which functions (if an) are the same. 7. f 8. f g 9 g 6 h 9 h 6 9. f f 5 g g 5 h 6 5. Graph the functions given b and and use the graphs to solve each inequalit. (a) < (b) > 5. Graph the functions given b and and use the graphs to solve each inequalit. (a) (b) > 5. Use a graphing utilit to graph e and each of the functions,,, and 5. Which function increases at the greatest rate as approaches? 5. Use the result of Eercise 5 to make a conjecture about the rate of growth of and n e, where n is a natural number and approaches. 55. Use the results of Eercises 5 and 5 to describe what is implied when it is stated that a quantit is growing eponentiall. 56. Which functions are eponential? (a) f (b) f (c) f (d) f Compound Interest In Eercises 57 60, complete the table to determine the balance A for P dollars invested at rate r for t ears and compounded n times per ear. n 65 Continuous A < h P $500, r %, t 0 ears 58. P $500, r.5%, t 0 ears 59. P $500, r %, t 0 ears 60. P $000, r 6%, t 0 ears Compound Interest In Eercises 6 6, complete the table to determine the balance A for $,000 invested at rate r for t ears, compounded continuousl. t A 6. r % 6. r 6% 6. r 6.5% 6. r.5% 65. Trust Fund On the da of a child s birth, a deposit of $0,000 is made in a trust fund that pas 5% interest, compounded continuousl. Determine the balance in this account on the child s 5th birthda. 66. Trust Fund A deposit of $5000 is made in a trust fund that pas 7.5% interest, compounded continuousl. It is specified that the balance will be given to the college from which the donor graduated after the mone has earned interest for 50 ears. How much will the college receive? 67. Inflation If the annual rate of inflation averages % over the net 0 ears, the approimate costs C of goods or services during an ear in that decade will be modeled b C t P.0 t, where t is the time in ears and P is the present cost. The price of an oil change for our car is presentl $.95. Estimate the price 0 ears from now. 68. Computer Virus The number V of computers infected b a computer virus increases according to the model V t 00e.605t, where t is the time in hours. Find the number of computers infected after (a) hour, (b).5 hours, and (c) hours. 69. Population Growth The projected populations of California for the ears 05 through 00 can be modeled b P.696e t, where P is the population (in millions) and t is the time (in ears), with t 5 corresponding to 05. (Source: U.S. Census Bureau) (a) Use a graphing utilit to graph the function for the ears 05 through 00. (b) Use the table feature of a graphing utilit to create a table of values for the same time period as in part (a). (c) According to the model, when will the population of California eceed 50 million? Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

10 80 Chapter 7 Eponential and Logarithmic Functions 70. Population The populations P (in millions) of Ital from 990 through 008 can be approimated b the model P 56.8e 0.005t, where t represents the ear, with t 0 corresponding to 990. (Source: U.S. Census Bureau, International Data Base) (a) According to the model, is the population of Ital increasing or decreasing? Eplain. (b) Find the populations of Ital in 000 and 008. (c) Use the model to predict the populations of Ital in 05 and Radioactive Deca Let Q represent a mass of radioactive plutonium 9 Pu (in grams), whose half-life is,00 ears. The quantit of plutonium present after t ears is Q 6 t,00. (a) Determine the initial quantit (when t 0). (b) Determine the quantit present after 75,000 ears. (c) Use a graphing utilit to graph the function over the interval t 0 to t 50, Radioactive Deca Let Q represent a mass of carbon C (in grams), whose half-life is 575 ears. The quantit of carbon present after t ears is Q 0 t 575. (a) Determine the initial quantit (when t 0). (b) Determine the quantit present after 000 ears. (c) Sketch the graph of this function over the interval t 0 to t 0, Depreciation After t ears, the value of a wheelchair conversion van that originall cost $0,500 7 depreciates so that each ear it is worth 8 of its value for the previous ear. (a) Find a model for V t, the value of the van after t ears. (b) Determine the value of the van ears after it was purchased. 7. Drug Concentration Immediatel following an injection, the concentration of a drug in the bloodstream is 00 milligrams per milliliter. After t hours, the concentration is 75% of the level of the previous hour. (a) Find a model for C t, the concentration of the drug after t hours. (b) Determine the concentration of the drug after 8 hours. True or False? In Eercises 75 and 76, determine whether the statement is true or false. Justif our answer. 75. The line is an asmptote for the graph of f e 7,80 99, Use a graphing utilit to graph each function. Use the graph to find where the function is increasing and decreasing, and approimate an relative maimum or minimum values. (a) f e (b) g 78. Graphical Analsis Use a graphing utilit to graph and e in the same viewing window. Using the trace feature, eplain what happens to the graph of as increases. 79. Graphical Analsis Use a graphing utilit to graph f 0.5 and in the same viewing window. What is the relationship between f and g as increases and decreases without bound? 80. Graphical Analsis Use a graphing utilit to graph each pair of functions in the same viewing window. Describe an similarities and differences in the graphs. (a), (b), 8. Compound Interest Use the formula A P r n nt g e 0.5 to calculate the balance of an account when P $000, r 6%, and t 0 ears, and compounding is done (a) b the da, (b) b the hour, (c) b the minute, and (d) b the second. Does increasing the number of compoundings per ear result in unlimited growth of the balance of the account? Eplain. CAPSTONE 8. The figure shows the graphs of, e, 0,, e, and 0. Match each function with its graph. [The graphs are labeled (a) through (f).] Eplain our reasoning. b a c d e f Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

11 7. Logarithmic Functions and Their Graphs 8 7. Logarithmic Functions and Their Graphs Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions. Recognize, evaluate, and graph natural logarithmic functions. Use logarithmic functions to model and solve real-life problems. Logarithmic Functions In Section.5, ou studied the concept of an inverse function. There, ou learned that if a function is one-to-one that is, if the function has the propert that no horizontal line intersects the graph of the function more than once the function must have an inverse function. B looking back at the graphs of the eponential functions introduced in Section 7., ou will see that ever function of the form f a passes the Horizontal Line Test and therefore must have an inverse function. This inverse function is called the logarithmic function with base a. DEFINITION OF LOGARITHMIC FUNCTION WITH BASE a Let a > 0 and a. For > 0, log if and onl if a a. The function given b f log a Read as log base a of. is called the logarithmic function with base a. The equations log and a a are equivalent. The first equation is in logarithmic form and the second is in eponential form. For eample, the logarithmic equation log 9 can be rewritten in eponential form as 9. The eponential equation 5 5 can be rewritten in logarithmic form as log 5 5. When evaluating logarithms, remember that a logarithm is an eponent. This means that log a is the eponent to which a must be raised to obtain. For instance, log 8 because must be raised to the third power to get 8. EXAMPLE Evaluating Logarithmic Functions Use the definition of logarithmic function to evaluate each function at the given value of. a. f log, b. f log, c. f log, d. f log 0, 00 Solution a. f log 5 because 5. b. f log 0 because 0. c. f log because. d. f 00 log 0 00 because Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

12 8 Chapter 7 Eponential and Logarithmic Functions The logarithmic function with base 0 is called the common logarithmic function. It is denoted b log 0 or simpl b log. On most calculators, this function is denoted b LOG. Eample shows how to use a calculator to evaluate common logarithmic functions. You will learn how to use a calculator to calculate logarithms to an base in the net section. EXAMPLE Evaluating the Common Logarithmic Function Use a calculator to evaluate the function given b f log at each value of. a. 0 b. c..5 d. Solution Function Value Graphing Calculator Kestrokes Displa a. f 0 log 0 LOG 0 ENTER b. LOG f log ENTER 0.77 c. f.5 log.5 LOG.5 ENTER d. f log LOG ENTER ERROR Note that the calculator displas an error message (or a comple number) when ou tr to evaluate log. The reason for this is that there is no real number power to which 0 can be raised to obtain. The following properties follow directl from the definition of the logarithmic function with base a. THEOREM 7. PROPERTIES OF LOGARITHMS. log because a 0 a 0.. log because a a a a.. log and a log a a a Inverse Properties. If log a log a, then. One-to-One Propert EXAMPLE Using Properties of Logarithms a. Solve the equation log log for. b. Solve the equation log for. c. Simplif the epression log 5 5. d. Simplif the epression 6 log 6 0. Solution a. Using the One-to-One Propert (Propert ), ou can conclude that. b. Using Propert, ou can conclude that. c. Using the Inverse Propert (Propert ), it follows that log 5 5. d. Using the Inverse Propert (Propert ), it follows that 6 log Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

13 7. Logarithmic Functions and Their Graphs 8 Graphs of Logarithmic Functions To sketch the graph of log a ou can use the fact that the graphs of inverse functions are reflections of each other in the line. EXAMPLE Graphs of Eponential and Logarithmic Functions In the same coordinate plane, sketch the graph of each function. 0 8 f() = = a. f b. g log Solution a. For f, construct a table of values. 6 g() = log 0 f 8 Figure B plotting these points and connecting them with a smooth curve, ou obtain the graph shown in Figure 7.9. b. Because g log is the inverse function of f, the graph of g is obtained b plotting the points f, and connecting them with a smooth curve. The graph of g is a reflection of the graph of f in the line, as shown in Figure 7.9. EXAMPLE 5 Sketching the Graph of a Logarithmic Function Sketch the graph of the common logarithmic function -intercept and vertical asmptote. f log. Identif the Solution Begin b constructing a table of values. Note that some of the values can be obtained without a calculator b using the Inverse Propert of Logarithms. Others require a calculator. Net, plot the points and connect them with a smooth curve, as shown in Figure 7.0. The -intercept of the graph is, 0 and the vertical asmptote is 0 ( -ais). Without calculator With calculator f log Vertical asmptote: = (, 0) f() = log 0 Figure 7.0 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

14 8 Chapter 7 Eponential and Logarithmic Functions The nature of the graph in Figure 7.0 is tpical of functions of the form f log a, a >. The have one -intercept and one vertical asmptote. Notice how slowl the graph rises for >. The basic characteristics of logarithmic graphs are summarized in Figure 7.. Figure 7. (, 0) = log a Graph of log a, a > Domain: 0, Range:, -intercept:, 0 Increasing One-to-one, therefore has an inverse function -ais is a vertical asmptote log as 0 a. Continuous Reflection of graph of a about the line The vertical asmptote occurs at 0, where log a is undefined. The basic characteristics of the graph of f a are shown below to illustrate the inverse relation between f a and g log a. Domain:, Range: 0, -intercept: 0, -ais is a horizontal asmptote a 0 as. In the net eample, the graph of log a is used to sketch the graphs of functions of the form f b ± log a c. Notice how a horizontal shift of the graph results in a horizontal shift of the vertical asmptote. EXAMPLE 6 Shifting Graphs of Logarithmic Functions STUDY TIP You can use our understanding of transformations to identif vertical asmptotes of logarithmic functions. For instance, in Eample 6(a), the graph of g f shifts the graph of f one unit to the right. So, the vertical asmptote of g is, one unit to the right of the vertical asmptote of the graph of f. The graph of each of the functions is similar to the graph of f log. a. Because g log f, the graph of g can be obtained b shifting the graph of f one unit to the right, as shown in Figure 7.(a). b. Because h log f, the graph of h can be obtained b shifting the graph of f two units upward, as shown in Figure 7.(b). (, ) f() = log h() = + log (, 0) (, 0) f() = log g() = log( ) (, 0) (a) Figure 7. (b) Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

15 7. Logarithmic Functions and Their Graphs 85 The Natural Logarithmic Function B looking back at the graph of the natural eponential function introduced in Section 7., ou will see that f e is one-to-one and so has an inverse function. This inverse function is called the natural logarithmic function and is denoted b the special smbol ln, read as the natural log of or el en of. Note that the natural logarithm is written without a base. The base is understood to be e. STUDY TIP Notice that as with ever other logarithmic function, the domain of the natural logarithmic function is the set of positive real numbers be sure ou see that ln is not defined for zero or for negative numbers. THEOREM 7. THE NATURAL LOGARITHMIC FUNCTION The function defined b f log e ln, > 0 is called the natural logarithmic function. (, e ) (0, ) f() = e ( (, e) (, 0) e, ) (e, ) = g() = f () = ln Reflection of graph of f e about the line. Figure 7. The definition above implies that the natural logarithmic function and the natural eponential function are inverse functions of each other. So, ever logarithmic equation can be written in an equivalent eponential form and ever eponential equation can be written in logarithmic form. That is, ln and e are equivalent equations. Because the functions given b f e and g ln are inverse functions of each other, their graphs are reflections of each other in the line. This reflective propert is illustrated in Figure 7.. The four properties of logarithms listed on page 8 are also valid for natural logarithms. THEOREM 7. PROPERTIES OF NATURAL LOGARITHMS. ln 0 because e 0.. ln e because e e.. ln e and e ln. Inverse Properties. If ln ln, then. One-to-One Propert EXAMPLE 7 Using Properties of Natural Logarithms Use the properties of natural logarithms to simplif each epression. ln a. ln b. e ln 5 c. d. ln e e. ln e f. e Solution a. ln Inverse Propert e ln e e ln b. e ln 5 5 Inverse Propert c. ln 0 0 Propert d. ln e ) Propert e. ln e Inverse Propert f. e ln Inverse Propert Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

16 86 Chapter 7 Eponential and Logarithmic Functions On most calculators, the natural logarithm is denoted b Eample 8. LN, as illustrated in EXAMPLE 8 Evaluating the Natural Logarithmic Function Use a calculator to evaluate the function given b f ln for each value of. a. b. 0. c. d. Solution Function Value a. f ln LN ENTER b. f 0. ln 0. LN. ENTER.0978 c. f ln LN ENTER ERROR d. LN ENTER f ln Graphing Calculator Kestrokes Displa In Eample 8, be sure ou see that ln gives an error message on most calculators. This occurs because the domain of ln is the set of positive real numbers (see Figure 7.). So, ln is undefined. NOTE Some graphing utilities displa a comple number instead of an ERROR message when evaluating an epression such as ln. EXAMPLE 9 Finding the Domains of Logarithmic Functions Find the domain of each function. a. f ln b. g ln c. h ln Solution a. Because ln is defined onl if > 0, it follows that the domain of f is,. The graph of f is shown in Figure 7.(a). b. Because ln is defined onl if > 0, it follows that the domain of g is,. The graph of g is shown in Figure 7.(b). c. Because ln is defined onl if > 0, it follows that the domain of h is all real numbers ecept 0. The graph of h is shown in Figure 7.(c). f() = ln( ) 5 g() = ln( ) h() = ln (a) (b) (c) Figure 7. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

17 7. Logarithmic Functions and Their Graphs 87 Application EXAMPLE 0 Human Memor Model Students participating in a pscholog eperiment attended several lectures on a subject and were given an eam. Ever month for a ear after the eam, the students were retested to see how much of the material the remembered. The average scores for the group are given b the human memor model f t 75 6 ln t, 0 t where t is the time in months. a. What was the average score on the original t 0 eam? b. What was the average score at the end of t months? c. What was the average score at the end of t 6 months? Algebraic Solution a. The original average score was f ln ln Substitute 0 for t. Simplif. Propert of natural logarithms Solution b. After months, the average score was f 75 6 ln Substitute for t ln Simplif Use a calculator Solution c. After 6 months, the average score was f ln 6 Substitute 6 for t ln 7 Simplif Use a calculator. 6.. Solution Graphical Solution Use a graphing utilit to graph the model 75 6 ln. Then use the value or trace feature to approimate the following. a. When 0, 75 (see Figure 7.5(a)). So, the original average score was 75. b. When, 68. (see Figure 7.5(b)). So, the average score after months was about 68.. c. When 6, 6. (see Figure 7.5(c)). So, the average score after 6 months was about (a) (b) (c) Figure 7.5 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

18 88 Chapter 7 Eponential and Logarithmic Functions 7. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 6, fill in the blanks.. The inverse function of the eponential function given b f a is called the function with base a.. The common logarithmic function has base.. The logarithmic function given b f ln is called the logarithmic function and has base.. The Inverse Properties of logarithms and eponentials state that log a a and. 5. The One-to-One Propert of natural logarithms states that if ln ln, then. 6. The domain of the natural logarithmic function is the set of. In Eercises 7, write the logarithmic equation in eponential form. For eample, the eponential form of log is log 7. log log log. log 5. log 6 8. log 6 8. log 8 In Eercises 5, write the eponential equation in logarithmic form. For eample, the logarithmic form of 8 is log 8. In Eercises 8, evaluate the function at the given value of without using a calculator. Function. f log. f log 5 5. f log 8 6. f log 7. g log a 8. g log b Value a b In Eercises 9, use a calculator to evaluate f log at the given value of. Round our result to three decimal places In Eercises 6, use the properties of logarithms to simplif the epression.. log 7. log log 9 5 In Eercises 7, find the domain, -intercept, and vertical asmptote of the logarithmic function and sketch its graph. 7. f log 8. g log 6 9. log 0. h log. f log 6. log 5. log. log In Eercises 5 50, use the graph of g log to match the given function with its graph. Then describe the relationship between the graphs of f and g. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (b) (c) (e) log 7 (d) 5. f log 6. f log 7. f log 8. f log 9. f log 50. f log (f) Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

19 7. Logarithmic Functions and Their Graphs 89 In Eercises 5 58, write the logarithmic equation in eponential form. 5. ln ln ln ln ln ln ln ln e In Eercises 59 66, write the eponential equation in logarithmic form. 59. e e e e e e e 66. e In Eercises 67 70, use a calculator to evaluate the function at the given value of. Round our result to three decimal places. Function 67. f ln 68. f ln 69. g 8 ln 70. g ln Value In Eercises 7 7, evaluate g ln at the given value of without using a calculator. 7. e 5 7. e 7. e e 5 In Eercises 75 78, find the domain, -intercept, and vertical asmptote of the logarithmic function and sketch its graph. 75. f ln 76. h ln g ln 78. f ln In Eercises 79 8, use a graphing utilit to graph the function. Be sure to use an appropriate viewing window. 79. f log f log 6 8. f ln 8. f ln 8. f ln 8 8. f ln In Eercises 85 9, use the One-to-One Propert to solve the equation for. 85. log 5 log log log log log log 5 log 89. ln ln 90. ln 7 ln 7 9. ln ln 9. ln ln 6 WRITING ABOUT CONCEPTS In Eercises 9 96, sketch the graphs of f and g and describe the relationship between the graphs of f and g. What is the relationship between the functions f and g? 9. f, g log 9. f 5, g log f e, g ln 96. f 8, g log Graphical Analsis Use a graphing utilit to graph f and g in the same viewing window and determine which is increasing at the greater rate as approaches. What can ou conclude about the rate of growth of the natural logarithmic function? (a) f ln, g (b) f ln, g 98. Compound Interest A principal P, invested at 5 % and compounded continuousl, increases to an amount K times the original principal after t ears, where t is given b t ln K (a) Complete the table and interpret our results. (b) Sketch a graph of the function. 99. Cable Television The numbers of cable television sstems C (in thousands) in the United States from 00 through 006 can be approimated b the model C t ln t, t 6, where t represents the ear, with t corresponding to 00. (Source: Warren Communication News) (a) Complete the table. (b) Use a graphing utilit to graph the function. (c) Can the model be used to predict the numbers of cable television sstems beond 006? Eplain. 00. Population The time t in ears for the world population to double if it is increasing at a continuous rate of r is given b t ln r. (a) Complete the table and interpret our results. r t K t t 5 6 C (b) Use a graphing utilit to graph the function. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

20 90 Chapter 7 Eponential and Logarithmic Functions 0. Human Memor Model Students in a mathematics class were given an eam and then retested monthl with an equivalent eam. The average scores for the class are given b the human memor model f t 80 7 log t, 0 t, where t is the time in months. (a) Use a graphing utilit to graph the model over the specified domain. (b) What was the average score on the original eam t 0? (c) What was the average score after months? (d) What was the average score after 0 months? 0. Sound Intensit The relationship between the number of decibels and the intensit of a sound I in watts per square meter is 0 log I 0. (a) Determine the number of decibels of a sound with an intensit of watt per square meter. (b) Determine the number of decibels of a sound with an intensit of watt per square meter. 0 (c) The intensit of the sound in part (a) is 00 times as great as that in part (b). Is the number of decibels 00 times as great? Eplain. 0. Monthl Pament The model t 6.65 ln > , approimates the length of a home mortgage of $50,000 at 6% in terms of the monthl pament. In the model, t is the length of the mortgage in ears and is the monthl pament in dollars. (a) Use the model to approimate the lengths of a $50,000 mortgage at 6% when the monthl pament is $897.7 and when the monthl pament is $659.. (b) Approimate the total amounts paid over the term of the mortgage with a monthl pament of $897.7 and with a monthl pament of $659.. (c) Approimate the total interest charges for a monthl pament of $897.7 and for a monthl pament of $659.. (d) What is the vertical asmptote for the model? Interpret its meaning in the contet of the problem. True or False? In Eercises 0 and 05, determine whether the statement is true or false. Justif our answer. 0. You can determine the graph of f log 6 b graphing g 6 and reflecting it about the -ais. 05. The graph of f log contains the point 7,. 06. Think About It Complete the table for f 0. 0 f Complete the table for f log. f Compare the two tables. What is the relationship between f 0 and f log? 07. (a) Complete the table for the function given b f ln f (b) Use the table in part (a) to determine what value f approaches as increases without bound. (c) Use a graphing utilit to confirm the result of part (b). CAPSTONE 08. The table of values was obtained b evaluating a function. Determine which of the statements ma be true and which must be false. (a) is an eponential function of. (b) is a logarithmic function of. 0 (c) is an eponential function of. (d) is a linear function of Writing Eplain wh log a is defined onl for 0 < a < and a >. In Eercises 0 and, (a) use a graphing utilit to graph the function, (b) use the graph to determine the intervals in which the function is increasing and decreasing, and (c) approimate an relative maimum or minimum values of the function. 0. f ln. h ln Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

21 7. Using Properties of Logarithms 9 7. Using Properties of Logarithms Use the change-of-base formula to rewrite and evaluate logarithmic epressions. Use properties of logarithms to evaluate or rewrite logarithmic epressions. Use properties of logarithms to epand or condense logarithmic epressions. Use logarithmic functions to model and solve real-life problems. Change of Base Most calculators have onl two tpes of log kes, one for common logarithms (base 0) and one for natural logarithms (base e). Although common logarithms and natural logarithms are the most frequentl used, ou ma occasionall need to evaluate logarithms with other bases. To do this, ou can use the following changeof-base formula. THEOREM 7.5 CHANGE-OF-BASE FORMULA Let a, b, and be positive real numbers such that a and b. Then log a can be converted to a different base as follows. Base b Base 0 Base e log a log b log b a log a log log a log a ln ln a One wa to look at the change-of-base formula is that logarithms with base a are simpl constant multiples of logarithms with base b. The constant multiplier is log b a. EXAMPLE Changing Bases Using Common Logarithms a. b. log 5 log 5 log log log log log a log log a Use a calculator. Simplif. EXAMPLE Changing Bases Using Natural Logarithms a. log 5 ln 5 ln log a ln ln a Use a calculator. Simplif. b. ln.89 log.5850 ln Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

22 9 Chapter 7 Eponential and Logarithmic Functions Properties of Logarithms You know from the preceding section that the logarithmic function with base a is the inverse function of the eponential function with base a. So, it makes sense that the properties of eponents should have corresponding properties involving logarithms. For instance, the eponential propert a 0 has the corresponding logarithmic propert log a 0. STUDY TIP There is no general propert that can be used to rewrite log a u ± v. Specificall, log a u v is not equal to log a u log a v. THEOREM 7.6 PROPERTIES OF LOGARITHMS Let a be a positive number such that a, and let n be a real number. If u and v are positive real numbers, the following properties are true. Logarithm with Base a Natural Logarithm. Product Propert: log a uv log a u log a v ln uv ln u ln v. Quotient Propert: log a u v log a u log a v ln u ln u ln v v. Power Propert: log ln u n a u n n log a u n ln u NOTE Pa attention to the domain when appling the properties of logarithms to a logarithmic function. For eample, the domain of f ln is all real 0, whereas the domain of g ln is all real > 0. A proof of the first propert listed above is given in Appendi A. EXAMPLE Using Properties of Logarithms The Granger Collection, New York JOHN NAPIER John Napier, a Scottish mathematician, developed logarithms as a wa to simplif some of the tedious calculations of his da. Beginning in 59, Napier worked about 0 ears on the invention of logarithms. Napier was onl partiall successful in his quest to simplif tedious calculations. Nonetheless, the development of logarithms was a step forward and received immediate recognition. Write each logarithm in terms of ln and ln. a. ln 6 b. ln 7 Solution a. ln 6 ln ln ln Rewrite 6 as. Product Propert b. ln ln ln 7 7 ln ln ln ln Quotient Propert Rewrite 7 as. Power Propert EXAMPLE Using Properties of Logarithms Find the eact value of each epression without using a calculator. a. log b. ln e 6 ln e 5 5 Solution a. log 5 5 log 5 5 log 5 5 b. ln e 6 ln e ln e6 e ln e ln e Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

23 7. Using Properties of Logarithms 9 STUDY TIP In Section 8., ou will see that properties of logarithms can also be used to rewrite logarithmic functions in forms that simplif the operations of calculus. Rewriting Logarithmic Epressions The properties of logarithms are useful for rewriting logarithmic epressions in forms that simplif the operations of algebra. This is true because these properties convert complicated products, quotients, and eponential forms into simpler sums, differences, and products, respectivel. EXAMPLE 5 Epanding Logarithmic Epressions Epand each logarithmic epression. a. b. log 5 5 ln 7 Solution a. log 5 log 5 log log log 5 log log Product Propert Power Propert b. 5 5 ln ln 7 7 ln 5 ln 7 Rewrite using rational eponent. Quotient Propert ln 5 ln 7 Power Propert In Eample 5, the properties of logarithms were used to epand logarithmic epressions. In Eample 6, this procedure is reversed and the properties of logarithms are used to condense logarithmic epressions. EXAMPLE 6 Condensing Logarithmic Epressions Condense each logarithmic epression. a. log log b. ln ln c. log log Solution a. Power Propert log log log log log Product Propert b. ln ln ln ln ln Power Propert Quotient Propert c. Product Propert log log log log log Power Propert Rewrite with a radical. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

24 9 Chapter 7 Eponential and Logarithmic Functions Application One method of determining how the - and -values for a set of nonlinear data are related is to take the natural logarithm of each of the - and -values. If these new points are graphed and fall on a line, then ou can determine that the - and -values are related b the equation ln m ln b where m is the slope of the line. Period (in ears) Mercur Figure 7.6 Venus Earth Mars Saturn Jupiter Mean distance (in astronomical units) EXAMPLE 7 Finding a Mathematical Model The table shows the mean distance from the sun and the period (the time it takes a planet to orbit the sun) for each of the si planets that are closest to the sun. In the table, the mean distance is given in terms of astronomical units (where Earth s mean distance is defined as.0), and the period is given in ears. Find an equation that relates and. Planet Mercur Venus Earth Mars Jupiter Saturn Mean distance, Period, Solution The points in the table above are plotted in Figure 7.6. From this figure it is not clear how to find an equation that relates and. To solve this problem, take the natural logarithm of each of the - and -values in the table. For instance, and ln 0.. ln Continuing this produces the following results. Venus Earth ln Mercur Figure 7.7 Mars Jupiter Saturn ln = ln ln Planet Mercur Venus Earth Mars Jupiter Saturn ln ln Now, b plotting the points in the second table, ou can see that all si of the points appear to lie in a line (see Figure 7.7). Using an two points, the slope of this line is found to be. You can therefore conclude that ln ln. The graph of this equation is shown in Figure 7.7. Using properties of logarithms, ou can solve for as shown below. ln ln Power Propert One-to-One Propert Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

25 7. Using Properties of Logarithms Eercises See for worked-out solutions to odd-numbered eercises. In Eercises, fill in the blanks.. To evaluate a logarithm to an base, ou can use the formula.. The change-of-base formula for base e is given b log a.. You can consider log a to be a constant multiple of log b ; the constant multiplier is.. The properties of logarithms are useful for logarithmic epressions in forms that simplif the operations of algebra. In Eercises 5, rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. 5. log log 7 7. log 5 8. log 9. log 0 0. log. log.6. log 7. In Eercises 0, evaluate the logarithm using the change-of-base formula. Round our result to three decimal places.. log 7. log 7 5. log 6. log 5 7. log log log log 0.05 In Eercises 6, use the properties of logarithms to rewrite and simplif the logarithmic epression.. log. log 8 9. log 5. log ln 5e 6. ln 6 6 e In Eercises 7, find the eact value of the logarithmic epression without using a calculator. (If this is not possible, state the reason.) log log 8 0. log 6 6. log 6. log 8. log. log 7 5. ln e.5 6. ln e log ln 8. ln e e 9. ln e ln e 5 0. ln e 6 ln e 5. log 5 75 log 5. log log In Eercises 6, use the properties of logarithms to epand the epression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.). ln. log 0z 5. log 6. log log 5 8. log 5 6 z 9. ln z 50. ln t 5. ln z 5. log ln ln z z, z >, > a log, a > 56. ln ln ln 59. ln 60. log z z 6. log 6. log 5 z 0 z 5 6. ln 6. ln In Eercises 65 8, condense the epression to the logarithm of a single quantit. 65. ln ln 66. ln ln t 67. log z log 68. log 5 8 log 5 t 69. log log 70. log 7 z 7. log 5 7. log 6 7. log log 7. ln 8 5 ln z 75. log log log z 76. log log log z 77. ln ln ln 78. ln z ln z 5 ln z ln ln ln 80. ln ln ln 8. log 8 log 8 log 8 8. log log 6 log Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

26 96 Chapter 7 Eponential and Logarithmic Functions In Eercises 8 and 8, compare the logarithmic quantities. If two are equal, eplain wh log log, log, log log log 7 70, log 7 5, Sound Intensit In Eercises 85 88, use the following information. The relationship between the number of decibels and the intensit of a sound I in watts per square meter is given b 0 log I 0. log Use the properties of logarithms to write the formula in simpler form, and determine the number of decibels of a sound with an intensit of 0 6 watt per square meter. 86. Find the difference in loudness between an average office with an intensit of watt per square meter and a broadcast studio with an intensit of watt per square meter. 87. Find the difference in loudness between a vacuum cleaner with an intensit of 0 watt per square meter and rustling leaves with an intensit of 0 watt per square meter. 88. You and our roommate are plaing our stereos at the same time and at the same intensit. How much louder is the music when both stereos are plaing compared with just one stereo plaing? Curve Fitting In Eercises 89 9, find a logarithmic equation that relates and. Eplain the steps used to find the equation WRITING ABOUT CONCEPTS In Eercises 9 and 9, use a graphing utilit to graph the two functions in the same viewing window. Use the graphs to verif that the epressions are equivalent. 9. f log 0 9. g ln ln 0 f ln g log 0 log 0 e 95. Sketch the graphs of f ln ln, g, h ln ln ln on the same set of aes. Which two functions have identical graphs? Eplain our reasoning. 96. Galloping Speeds of Animals Four-legged animals run with two different tpes of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times, whereas an animal that is galloping has all four feet off the ground at some point in its stride. The number of strides per minute at which an animal breaks from a trot to a gallop depends on the weight of the animal. Use the table to find a logarithmic equation that relates an animal s weight (in pounds) and its lowest galloping speed (in strides per minute). Weight, Galloping speed, Weight, Galloping speed, Nail Length The approimate lengths and diameters (in inches) of common nails are shown in the table. Find a logarithmic equation that relates the diameter of a common nail to its length. Length, Diameter, Length, 5 6 Diameter, Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

27 7. Using Properties of Logarithms Comparing Models A cup of water at an initial temperature of 78 C is placed in a room at a constant temperature of C. The temperature of the water is measured ever 5 minutes during a half-hour period. The results are recorded as ordered pairs of the form t, T, where t is the time (in minutes) and T is the temperature (in degrees Celsius). 0, 78.0, 5, 66.0, 0, 57.5, 5, 5., 0, 6., 5,., 0, 9.6 (a) The graph of the model for the data should be asmptotic with the graph of the temperature of the room. Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utilit to plot the data points t, T and t, T. (b) An eponential model for the data t, T is given b T t. Solve for T and graph the model. Compare the result with the plot of the original data. (c) Take the natural logarithms of the revised temperatures. Use a graphing utilit to plot the points t, ln T and observe that the points appear to be linear. Use the regression feature of the graphing utilit to fit a line to these data. This resulting line has the form ln T at b. Solve for T, and verif that the result is equivalent to the model in part (b). (d) Fit a rational model to the data. Take the reciprocals of the -coordinates of the revised data points to generate the points t, T. Use a graphing utilit to graph these points and observe that the appear to be linear. Use the regression feature of a graphing utilit to fit a line to these data. The resulting line has the form at b. T Solve for T, and use a graphing utilit to graph the rational function and the original data points. (e) Wh did taking the logarithms of the temperatures lead to a linear scatter plot? Wh did taking the reciprocals of the temperatures lead to a linear scatter plot? True or False? In Eercises 99 0, determine whether the statement is true or false given that f ln. Justif our answer. 99. f f a f a f, a > 0, > 0 0. f f f, > 0. f f 0. If f u f v, then v u. 0. If f < 0, then 0 < <. In Eercises 05 0, use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utilit to graph the ratio. 05. f log 06. f log 07. f log 08. f log 09. f log.8 0. f log.. Graphical Analsis Use a graphing utilit to graph the functions given b ln ln and ln in the same viewing window. Does the graphing utilit show the functions with the same domain? If so, should it? Eplain our reasoning. CAPSTONE. A classmate claims that the following are true. (a) ln u v ln u ln v ln uv (b) ln u v ln u ln v ln u v (c) ln u n n ln u ln u n Discuss how ou would demonstrate that these claims are not true.. Proof Prove that log u b v log b u log b v.. Proof Prove that log b u n n log b u. 5. Think About It For how man integers between and 0 can the natural logarithms be approimated given the values ln 0.69, ln.0986, and ln 5.609? Approimate these logarithms (do not use a calculator). Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

28 98 Chapter 7 Eponential and Logarithmic Functions 7. Eponential and Logarithmic Equations Solve simple eponential and logarithmic equations. Solve more complicated eponential equations. Solve more complicated logarithmic equations. Use eponential and logarithmic equations to model and solve real-life problems. Introduction So far in this chapter, ou have studied the definitions, graphs, and properties of eponential and logarithmic functions. In this section, ou will stud procedures for solving equations involving these eponential and logarithmic functions. There are two basic strategies for solving eponential or logarithmic equations. The first is based on the One-to-One Properties and the second is based on the Inverse Properties. For a > 0 and a, the following properties are true for all and for which log a and log a are defined. One-to-One Properties a a if and onl if. log a log a if and onl if. Inverse Properties a log a log a a EXAMPLE Solving Simple Equations Original Equation Rewritten Equation Solution Propert a. 5 5 One-to-One b. ln ln 0 ln ln One-to-One 9 c. One-to-One d. e 7 ln e ln 7 ln 7 Inverse e. ln e ln e e Inverse f. log 0 log 0 0 Inverse g. log log 8 Inverse 0 The strategies used in Eample are summarized as follows. STRATEGIES FOR SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS. Rewrite the original equation in a form that allows the use of the One-to-One Properties of eponential or logarithmic functions.. Rewrite an eponential equation in logarithmic form and appl the Inverse Propert of logarithmic functions.. Rewrite a logarithmic equation in eponential form and appl the Inverse Propert of eponential functions. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

29 7. Eponential and Logarithmic Equations 99 Solving Eponential Equations EXAMPLE Solving Eponential Equations STUDY TIP Another wa to solve Eample (b) is b taking the natural log of each side and then appling the Power Propert, as follows. ln ln ln ln ln ln.807 As ou can see, ou obtain the same result as in Eample (b). Solve each equation and approimate the result to three decimal places, if necessar. a. b. e e Solution a. e e Write original equation. One-to-One Propert Write in general form. Factor. Set st factor equal to 0. Set nd factor equal to 0. The solutions are and. Check these in the original equation. b. log log log Write original equation. Divide each side b. Take log (base ) of each side. Inverse Propert ln ln.807 Change-of-base formula Use a calculator. The solution is log.807. Check this in the original equation. In Eample (b), the eact solution is log and the approimate solution is.807. An eact answer is preferred when the solution is an intermediate step in a larger problem. For a final answer, an approimate solution is easier to comprehend. EXAMPLE Solving an Eponential Equation STUDY TIP When taking the logarithm of each side of an eponential equation, choose the base for the logarithm to be the same as the base in the eponential equation. In Eample (b), base was chosen, and in Eample, base e was chosen for the logarithm. Solve e 5 60 and approimate the result to three decimal places. Solution e 5 60 e 55 ln e ln 55 ln Write original equation. Subtract 5 from each side. Take natural log of each side. Inverse Propert Use a calculator. The solution is ln Check this in the original equation. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

30 500 Chapter 7 Eponential and Logarithmic Functions EXAMPLE Solving an Eponential Equation Solve t 5 and approimate the result to three decimal places. Solution t 5 t 5 5 t 5 5 Write original equation. Add to each side. Divide each side b. log t 5 log 5 Take log (base ) of each side. t 5 log 5 t 5 log 7.5 t 5 log 7.5 t.7 Inverse Propert Add 5 to each side. Divide each side b. Use a calculator. The solution is t 5 log Check this in the original equation. When an equation involves two or more eponential epressions, ou can still use a procedure similar to that demonstrated in Eamples,, and. However, the algebra is a bit more complicated. In such cases, remember that a graph can help ou check the reasonableness of our solution. EXAMPLE 5 Solving an Eponential Equation of Quadratic Tpe Solve e e 0. Algebraic Solution e e 0 Write original equation. e e 0 Write in quadratic form. e e 0 Factor. e 0 Set st factor equal to 0. ln Solution e 0 Set nd factor equal to 0. 0 Solution The solutions are ln 0.69 and 0. Check these in the original equation. Graphical Solution Use a graphing utilit to graph e e. Use the zero or root feature or the zoom and trace features of the graphing utilit to approimate the values of for which 0. In Figure 7.8, ou can see that the zeros occur at 0 and at So, the solutions are 0 and = e e + Figure 7.8 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

31 7. Eponential and Logarithmic Equations 50 Solving Logarithmic Equations To solve a logarithmic equation, ou can write it in eponential form. ln e ln e e Logarithmic form Eponentiate each side. Eponential form This procedure is called eponentiating each side of an equation. EXAMPLE 6 Solving Logarithmic Equations STUDY TIP Remember to check our solutions in the original equation when solving equations to verif that the answer is correct and to make sure that the answer lies in the domain of the original equation. a. ln e ln e e Original equation Eponentiate each side. Inverse Propert b. log 5 log Original equation One-to-One Propert Add and to each side. Divide each side b. c. log 6 log 6 5 log 6 Original equation log 6 5 log Quotient Propert of Logarithms One-to-One Propert Cross multipl. Isolate. Divide each side b 7. EXAMPLE 7 Solving a Logarithmic Equation Solve 5 ln and approimate the result to three decimal places. Algebraic Solution 5 ln Write original equation. ln Subtract 5 from each side. ln Divide each side b. Graphical Solution Use a graphing utilit to graph 5 ln and in the same viewing window. Use the intersect feature or the zoom and trace features to approimate the intersection point, as shown in Figure 7.9. So, the solution is e ln e Eponentiate each side. e Inverse Propert Use a calculator. The solution is e Check this in the original equation = (e /, ) = 5 + ln Figure 7.9 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

32 50 Chapter 7 Eponential and Logarithmic Functions EXAMPLE 8 Solving a Logarithmic Equation STUDY TIP Notice in Eample 9 that the logarithmic part of the equation is condensed into a single logarithm before eponentiating each side of the equation. Solve log 5. Solution log 5 Write original equation. log 5 Divide each side b. 5 log 5 5 Eponentiate each side (base 5). 5 Inverse Propert 5 Divide each side b. The solution is 5. Check this in the original equation. Because the domain of a logarithmic function generall does not include all real numbers, ou should be sure to check for etraneous solutions of logarithmic equations. EXAMPLE 9 Checking for Etraneous Solutions Solve log 5 log. Algebraic Solution log 5 log Write original equation. log 5 Product Propert of Logarithms 0 log Eponentiate each side (base 0) Inverse Propert 0 0 Write in general form. 5 0 Factor. 5 0 Set st factor equal to 0. 5 Solution 0 Set nd factor equal to 0. Solution The solutions appear to be 5 and. However, when ou check these in the original equation, ou can see that 5 is the onl solution. Graphical Solution Use a graphing utilit to graph and in the same viewing window. From the graph shown in Figure 7.0, it appears that the graphs intersect at one point. Use the intersect feature or the zoom and trace features to determine that the graphs intersect at approimatel 5,. So, the solution is 5. Verif that 5 is an eact solution algebraicall. 5 log 5 log = log 5 + log( ) 0 = 9 Figure 7.0 In Eample 9, the domain of log 5 is > 0 and the domain of log is >, so the domain of the original equation is >. Because the domain is all real numbers greater than, the solution is etraneous. The graph in Figure 7.0 verifies this conclusion. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

33 7. Eponential and Logarithmic Equations 50 Applications EXAMPLE 0 Doubling an Investment You have deposited $500 in an account that pas 6.75% interest, compounded continuousl. How long will it take our mone to double? Solution Using the formula for continuous compounding, ou can find that the balance in the account is A Pe rt A 500e t. To find the time required for the balance to double, let A 000 and solve the resulting equation for t. 500e t 000 Substitute 000 for A. e t Divide each side b 500. ln e t ln Take natural log of each side t ln Inverse Propert t ln Divide each side b t 0.7 Use a calculator. The balance in the account will double after approimatel 0.7 ears. This result is demonstrated graphicall in Figure 7.. A 00 (0.7, 000) Account balance (in dollars) A = 500e t 500 (0, 500) Time (in ears) Figure 7. t In Eample 0, an approimate answer of 0.7 ears is given. Within the contet of the problem, the eact solution, ln ears does not make sense as an answer. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

34 50 Chapter 7 Eponential and Logarithmic Functions EXAMPLE Retail Sales The retail sales (in billions) of e-commerce companies in the United States from 00 through 007 can be modeled b ln t, t 7 where t represents the ear, with t corresponding to 00 (see Figure 7.). During which ear did the sales reach $08 billion? (Source: U.S. Census Bureau) Sales (in billions) t Year ( 00) Figure 7. Solution ln t Write original equation ln t 08 Substitute 08 for. 6.7 ln t 657 Add 59 to each side. ln t 657 Divide each side b e ln t e Eponentiate each side. t e Inverse Propert t 6 Use a calculator. The solution is t 6. Because t represents 00, it follows that the sales reached $08 billion in Eercises See for worked-out solutions to odd-numbered eercises. In Eercises, fill in the blanks.. To an equation in means to find all values of for which the equation is true.. To solve eponential and logarithmic equations, ou can use the following One-to-One and Inverse Properties. (a) a a if and onl if. (b) log a log a if and onl if. (c) a log a (d) log a a. To solve eponential and logarithmic equations, ou can use the following strategies. (a) Rewrite the original equation in a form that allows the use of the Properties of eponential or logarithmic functions. (b) Rewrite an eponential equation in form and appl the Inverse Propert of functions. (c) Rewrite a logarithmic equation in form and appl the Inverse Propert of functions.. An solution does not satisf the original equation. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

35 7. Eponential and Logarithmic Equations 505 In Eercises 5, determine whether each -value is a solution (or an approimate solution) of the equation. e (a) 5 (a) (b) (b) 7. e (a) e 5 (a) ln 5 (b) ln 5 (b).708 (c).9 (c) ln 6 9. log 0. log 0 (a). (a) 0 (b) (b) 7 (c) 6 (c) 0. ln 5.8 (a) (b) (c) ln 5.8 e ln.8 (a) (b) (c) e ln.8 In Eercises, solve for ln ln 0 8. ln ln e e ln log log. log 5 In Eercises 5 8, approimate the point of intersection of the graphs of f and g. Then solve the equation f g algebraicall to verif our approimation. 5. f 6. f 7 g 8 g 9 8 f g f g 8 7. f log 8. f ln g g 0 In Eercises 9 70, solve the eponential equation algebraicall. Approimate the result to three decimal places. 9. e e 0. e e 8. e e. e e e 0 6. e 9 7. e t 0.0. t e 5. e e e e e e e e 5e e e 0 6. e 9e e 0 e e e f g t t t t 0 In Eercises 7 80, use a graphing utilit to graph and solve the equation. Approimate the result to three decimal places. Verif our result algebraicall e 5 7. e e e 77. e 0.09t 78. e e 0.5t e g f Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

36 506 Chapter 7 Eponential and Logarithmic Functions In Eercises 8, solve the logarithmic equation algebraicall. Approimate the result to three decimal places. 8. ln 8. ln.6 8. ln ln ln ln log log z 89. ln ln 7 9. ln 9. ln ln 5 6 ln 0 ln 7 ln 6 log 0.5 log 6 ln ln ln ln ln ln ln ln 0. ln 5 ln ln ln ln ln log log log log 0 log log log log log log 6 log log 0. log log 8. log 8 log. log log In Eercises 6, use a graphing utilit to graph and solve the equation. Approimate the result to three decimal places. Verif our result algebraicall.. ln 0. 0 ln 0 5. ln 6. ln ln In Eercises 7, solve the equation algebraicall. Round the result to three decimal places. Verif our answer using a graphing utilit. 7. e e 0 8. e e 0 9. e e 0 0. e e 0. ln 0. ln 0 ln. 0. ln 0 WRITING ABOUT CONCEPTS 5. Finance You are investing P dollars at an annual interest rate of r, compounded continuousl, for t ears. Which of the following would result in the highest value of the investment? Eplain our reasoning. (a) Double the amount ou invest. (b) Double our interest rate. (c) Double the number of ears. 6. Write a paragraph eplaining whether the time required for an investment to double depends on the size of the investment. Compound Interest In Eercises 7 0, $500 is invested in an account at interest rate r, compounded continuousl. Find the time required for the amount to (a) double and (b) triple. 7. r r r r Think About It Are the times required for the investments in Eercises 7 0 to quadruple twice as long as the times for them to double? Give a reason for our answer and verif our answer algebraicall.. Demand The demand equation for a limited edition coin set is p 000 Find the demand for a price of (a) p $9.50 and (b) p $ Demand The demand equation for a hand-held electronic organizer is p e e Find the demand for a price of (a) p $600 and (b) p $00.. Forest Yield The ield V (in millions of cubic feet per acre) for a forest at age t ears is given b V 6.7e 8. t. (a) Use a graphing utilit to graph the function. (b) Determine the horizontal asmptote of the function. Interpret its meaning in the contet of the problem. (c) Find the time necessar to obtain a ield of. million cubic feet. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

37 7. Eponential and Logarithmic Equations Trees per Acre The number N of trees of a given species per acre is approimated b the model N , 5 0, where is the average diameter of the trees (in inches) feet above the ground. Use the model to approimate the average diameter of the trees in a test plot when N. 6. U.S. Currenc The values (in billions of dollars) of U.S. currenc in circulation in the ears 000 through 007 can be modeled b 5 ln t, 0 t 7, where t represents the ear, with t 0 corresponding to 000. During which ear did the value of U.S. currenc in circulation eceed $690 billion? (Source: Board of Governors of the Federal Reserve Sstem) 7. Medicine The numbers of freestanding ambulator care surger centers in the United States from 000 through 007 can be modeled b 875 where t represents the ear, with t 0 corresponding to 000. (Source: Verispan) (a) Use a graphing utilit to graph the model. (b) Use the trace feature of the graphing utilit to estimate the ear in which the number of surger centers eceeded Average Heights The percent m of American males between the ages of 8 and who are no more than inches tall is modeled b m and the percent f of American females between the ages of 8 and who are no more than inches tall is modeled b f (Source: U.S. National Center for Health Statistics) (a) Use the graph to determine an horizontal asmptotes of the graphs of the functions. Interpret the meaning in the contet of the problem. Percent of population 00 e e e 0.808t, 0 t 7 f() m() Height (in inches) (b) What is the average height of each se? 9. Learning Curve In a group project in learning theor, a mathematical model for the proportion P of correct responses after n trials was found to be P 0.8 e 0.n. (a) Use a graphing utilit to graph the function. (b) Use the graph to determine an horizontal asmptotes of the graph of the function. Interpret the meaning of the upper asmptote in the contet of this problem. (c) After how man trials will 60% of the responses be correct? 0. Automobiles Automobiles are designed with crumple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer g s the crash victims eperience. (One g is equal to the acceleration due to gravit. For ver short periods of time, humans have withstood as much as 0 g s.) In crash tests with vehicles moving at 90 kilometers per hour, analsts measured the numbers of g s eperienced during deceleration b crash dummies that were permitted to move meters during impact. The data are shown in the table. A model for the data is given b ln 6.9 where is the number of g s g s (a) Complete the table using the model (b) Use a graphing utilit to graph the data points and the model in the same viewing window. How do the compare? (c) Use the model to estimate the distance traveled during impact if the passenger deceleration must not eceed 0 g s. (d) Do ou think it is practical to lower the number of g s eperienced during impact to fewer than? Eplain our reasoning. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

38 508 Chapter 7 Eponential and Logarithmic Functions. Data Analsis An object at a temperature of 60 C was removed from a furnace and placed in a room at 0 C. The temperature T of the object was measured each hour h and recorded in the table. A model for the data is given b T 0 7 h. The graph of this model is shown in the figure. Hour, h 0 5 Temperature, T (a) Use the graph to identif the horizontal asmptote of the model and interpret the asmptote in the contet of the problem. (b) Use the model to approimate the time when the temperature of the object was 00 C. Temperature (in degrees Celsius) T Time (in hours). Data Analsis The personal consumption medical care ependitures E (in billions of dollars) for selected ears from 960 to 000 are shown in the table. t E A model for these data is E 0.9e 0.066t, where t is the time in ears, with t 0 corresponding to 960. (Source: U.S. Bureau of Economic Analsis) (a) Use a graphing utilit to graph the data points and the model in the same viewing window. How do the compare? (b) Use the model to estimate the personal consumption medical ependitures for 00, 05, and 00. (c) Algebraicall find the ear, according to the model, when personal consumption medical ependitures eceed trillion dollars. (d) Do ou believe that the future personal consumption medical ependitures can be predicted using the given model? Eplain our reasoning. h True or False? In Eercises 6, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justif our answer.. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.. The logarithm of the sum of two numbers is equal to the product of the logarithms of the numbers. 5. The logarithm of the difference of two numbers is equal to the difference of the logarithms of the numbers. 6. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers. 7. Think About It Is it possible for a logarithmic equation to have more than one etraneous solution? Eplain. 8. The effective ield of a savings plan is the percent increase in the balance after ear. Find the effective ield for each savings plan when $000 is deposited in a savings account. Which savings plan has the greatest effective ield? Which savings plan will have the highest balance after 5 ears? (a) 7% annual interest rate, compounded annuall (b) 7% annual interest rate, compounded continuousl (c) 7% annual interest rate, compounded quarterl (d) 7.5% annual interest rate, compounded quarterl 9. Graphical Analsis Let f log a and g a where a >. (a) Let a. and use a graphing utilit to graph the two functions in the same viewing window. What do ou observe? Approimate an points of intersection of the two graphs. (b) Determine the value(s) of a for which the two graphs have one point of intersection. (c) Determine the value(s) of a for which the two graphs have two points of intersection. CAPSTONE 50. Write two or three sentences stating the general guidelines that ou follow when solving (a) eponential equations and (b) logarithmic equations. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

39 7.5 Eponential and Logarithmic Models Eponential and Logarithmic Models Recognize the five most common tpes of models involving eponential and logarithmic functions. Use eponential growth and deca functions to model and solve real-life problems. Use Gaussian functions to model and solve real-life problems. Use logistic growth functions to model and solve real-life problems. Use logarithmic functions to model and solve real-life problems. STUDY TIP The models for eponential growth and deca var onl in the sign of the real number b. Introduction The five most common tpes of mathematical models involving eponential functions and logarithmic functions are as follows.. Eponential growth model: ae b, a > 0, b > 0. Eponential deca model: ae b, a > 0, b > 0. Gaussian model: ae ( b) c, a > 0 a. Logistic growth model: be r, a > 0 5. Logarithmic models: a b ln, a b log The basic shapes of the graphs of these functions are shown in Figure 7.. = e = e = e Eponential growth model Eponential deca model Gaussian model = + ln = + log = + e 5 Logistic growth model Natural logarithmic model Common logarithmic model Figure 7. You can often gain quite a bit of insight into a situation modeled b an eponential or logarithmic function b identifing and interpreting the function s asmptotes. Use the graphs in Figure 7. to identif the asmptotes of the graph of each function. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

40 50 Chapter 7 Eponential and Logarithmic Functions Dollars (in billions) S Eponential Growth and Deca EXAMPLE Online Advertising Estimates of the amounts (in billions of dollars) of U.S. online advertising spending from 007 through 0 are shown in the table. A scatter plot of the data is shown in Figure 7.. (Source: emarketer) Figure Year (7 007) t Year Advertising spending An eponential growth model that approimates these data is given b S 0.e 0.0t, 7 t Algebraic Solution The following table compares the two sets of advertising spending figures. Year Advertising spending Model where S is the amount of spending (in billions) and t 7 represents 007. Compare the values given b the model with the estimates shown in the table. According to this model, when will the amount of U.S. online advertising spending reach $0 billion? Graphical Solution Use a graphing utilit to graph the model 0.e 0.0 and the data in the same viewing window. You can see in Figure 7.5 that the model appears to fit the data closel. 50 To find when the amount of U.S. online advertising spending will reach $0 billion, let S 0 in the model and solve for t. 0.e 0.0t S 0.e 0.0t 0 e 0.0t.87 ln e 0.0t ln t.58 t. Write original model. Substitute 0 for S. Divide each side b 0.. Take natural log of each side. Inverse Propert Divide each side b 0.0. According to the model, the amount of U.S. online advertising spending will reach $0 billion in Figure 7.5 Use the zoom and trace features of the graphing utilit to find that the approimate value of for 0 is.. So, according to the model, the amount of U.S. online advertising spending will reach $0 billion in 0. TECHNOLOGY Some graphing utilities have an eponential regression feature that can be used to find eponential models that represent data. If ou have such a graphing utilit, tr using it to find an eponential model for the data given in Eample. How does our model compare with the model given in Eample? Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

41 7.5 Eponential and Logarithmic Models 5 In Eample, ou were given the eponential growth model. But suppose this model were not given. How could ou find such a model? One technique for doing this is demonstrated in Eample. EXAMPLE Modeling Population Growth In a research eperiment, a population of fruit flies is increasing according to the law of eponential growth. After das there are 00 flies, and after das there are 00 flies. How man flies will there be after 5 das? Solution Let be the number of flies at time t. From the given information, ou know that 00 when t and 00 when t. Substituting this information into the model ae bt produces 00 ae b and 00 ae b. To solve for b, solve for a in the first equation. 00 ae b a 00 e b Solve for a in the first equation. Then substitute the result into the second equation. Population (, 00) = e 0.59t (, 00) (5, 5) 5 Time (in das) Figure 7.6 t 00 ae b e b eb 00 eb 00 ln b ln b Write second equation. Substitute Divide each side b 00. Take natural log of each side. Solve for b. Using b ln and the equation ou found for a, ou can determine that a 00 e ln So, with a and b ln 0.59, the eponential growth model is e 0.59t 00 e ln for a. e b as shown in Figure 7.6. This implies that, after 5 das, the population will be R e flies. Ratio 0 (0 ) t = 0 t = 5700 t = 9,000 In living organic material, the ratio of the number of radioactive carbon isotopes (carbon ) to the number of nonradioactive carbon isotopes (carbon ) is about to 0. When organic material dies, its carbon content remains fied, whereas its radioactive carbon begins to deca with a half-life of about 5700 ears. To estimate the age of dead organic material, scientists use the following formula, which denotes the ratio of carbon to carbon present at an time t (in ears). 0 Figure ,000 t R 8 0e t Carbon dating model The graph of R is shown in Figure 7.7. Note that R decreases as t increases. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

42 5 Chapter 7 Eponential and Logarithmic Functions Gaussian Models As mentioned at the beginning of this section, Gaussian models are of the form ae b c. This tpe of model is commonl used in probabilit and statistics to represent populations that are normall distributed. For standard normal distributions, the model takes the form e where is the standard deviation ( is the lowercase Greek letter sigma). The graph of a Gaussian model is called a bell-shaped curve. Tr to sketch the standard normal distribution curve with a graphing utilit. Can ou see wh it is called a bell-shaped curve? The average value for a population can be found from the bell-shaped curve b observing where the maimum -value of the function occurs. The -value corresponding to the maimum -value of the function represents the average value of the independent variable in this case,. EXAMPLE SAT Scores In 008, the Scholastic Aptitude Test (SAT) math scores for college-bound seniors roughl followed the normal distribution given b 0.00e 55 6,9, where is the SAT score for mathematics. Sketch the graph of this function. From the graph, estimate the average SAT score. (Source: College Board) Solution The graph of the function is shown in Figure 7.8. On this bell-shaped curve, the maimum value of the curve represents the average score. From the graph, ou can estimate that the average mathematics score for college-bound seniors in 008 was 55. Distribution = 55 50% of population Figure Score Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

43 7.5 Eponential and Logarithmic Models 5 Decreasing rate of growth Increasing rate of growth Logistic Growth Models Some populations initiall have rapid growth, followed b a declining rate of growth, as indicated b the graph in Figure 7.9. One model for describing this tpe of growth pattern is the logistic curve given b the function a be r where is the population size and is the time. An eample is a bacteria culture that is initiall allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. A logistic growth curve is also called a sigmoidal curve. Figure 7.9 EXAMPLE Spread of a Virus On a college campus of 5000 students, one student returns from vacation with a contagious and long-lasting flu virus. The spread of the virus is modeled b 5000, t 999e 0.8t 0 where is the total number of students infected after t das. The college will cancel classes when 0% or more of the students are infected. a. How man students are infected after 5 das? b. After how man das will the college cancel classes? Algebraic Solution a. After 5 das, the number of students infected is e b. Classes are canceled when the number infected is e 0.8t 999e 0.8t.5 e 0.8t ln e 0.8t ln t ln t.5 ln t e 5. So, after about 0 das, at least 0% of the students will be infected, and the college will cancel classes. Graphical Solution 5000 a. Use a graphing utilit to graph Use 999e 0.8. the value feature or the zoom and trace features of the graphing utilit to estimate that 5 when 5. So, after 5 das, about 5 students will be infected. b. Classes are canceled when the number of infected students is Use a graphing utilit to graph and in the same viewing window. Use the intersect feature or the zoom and trace features of the graphing utilit to find the point of intersection of the graphs. In Figure 7.0, ou can see that the point of intersection occurs near 0.. So, after about 0 das, at least 0% of the students will be infected, and the college will cancel classes = e 0.8 = Figure e 0.8 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

44 5 Chapter 7 Eponential and Logarithmic Functions CLARO CORTES IV/Reuters /Landov On Ma, 008, an earthquake of magnitude 7.9 struck Eastern Sichuan Province, China. The total economic loss was estimated at 86 billion U.S. dollars. Logarithmic Models EXAMPLE 5 Magnitudes of Earthquakes On the Richter scale, the magnitude R of an earthquake of intensit I is given b R log I I 0 where I 0 is the minimum intensit used for comparison. Find the intensit of each earthquake. (Intensit is a measure of the wave energ of an earthquake.) a. Nevada in 008: R 6.0 b. Eastern Sichuan, China in 008: R 7.9 c. Offshore Maule, Chile in 00: R 8.8 Solution a. Because I 0 and R 6.0, ou have 6.0 log I log I I 0 6.0,000,000. b. For R 7.9, ou have 7.9 log I log I I ,00,000. c. For R 8.8, ou have 8.8 log I Substitute for Eponentiate each side. Inverse Propert Substitute for Eponentiate each side. Inverse Propert Substitute for I 0 I 0 I 0 and 6.0 for R. and 7.9 for R. and 8.8 for R log I Eponentiate each side. I Inverse Propert 6,000,000. Note that an increase of.9 units on the Richter scale (from 6.0 to 7.9) represents an increase in intensit b a factor of 79,00,000,000, In other words, the intensit of the earthquake in Eastern Sichuan was about 79 times as great as that of the earthquake in Nevada. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it