0 Chapter Eponential and Logarithmic Functions (, e) f() = e (, e ) (0, ) (, e ) FIGURE.9 The Natural Base e In man applications, the most convenient choice for a base is the irrational number e.78888.... 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.9. Be sure ou see that for the eponential function f e, e is the constant.78888..., whereas is the variable. Eample 6 Evaluating the Natural Eponential Function Use a calculator to evaluate the function given b f e at each indicated value of. a. b. c. 0.5 d. 0. 8 7 f() = e 0. 6 5 FIGURE.0 8 7 6 5 g() = e 0.58 FIGURE. Function Value Graphing Calculator Kestrokes Displa a. f e e ENTER 0.55 b. f e e ENTER 0.67879 c. f 0.5 e 0.5 e 0.5 ENTER.805 d. f 0. e 0. e 0. ENTER 0.7088 Now tr Eercise. Graphing Natural Eponential Functions Sketch the graph of each natural eponential function. a. b. Eample 7 f e 0. g e 0.58 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 Figures.0 and.. Note that the graph in Figure.0 is increasing, whereas the graph in Figure. is decreasing. 0 f 0.97.8.57.000.5..09 g.89.595 0.89 0.500 0.80 0.57 0.088 Now tr Eercise.
Section. Eponential Functions and Their Graphs Applications One of the most familiar eamples of eponential growth is an investment earning continuousl compounded interest. Using eponential functions, ou can develop a formula for interest compounded n times per ear and show how it leads to continuous compounding. Suppose a principal P is invested at an annual interest rate r, compounded once per ear. If the interest is added to the principal at the end of the ear, the new balance is P This pattern of multipling the previous principal b r is then repeated each successive ear, as shown below. Year 0 P P Pr P r. P P P P r Balance After Each Compounding P P r P r r P r P.. P r P r r P r. t 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 A P r n nt. Amount (balance) with n compoundings per ear m 0 00,000 0,000 00,000,000,000 0,000,000 m m.5976.70889.7699.78597.78687.788069.78869 e 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 P m m rt. Amount with n compoundings per ear Substitute mr for n. Simplif. Propert of eponents As m increases without bound, the table at the left shows that m m e as m. From this, ou can conclude that the formula for continuous compounding is A Pe rt. Substitute e for m m.
Chapter Eponential and Logarithmic Functions WARNING / CAUTION Be sure ou see that the annual interest rate must be written in decimal form. For instance, 6% should be written as 0.06. 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 Eample 8 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. continuousl. a. For quarterl compounding, ou have n. So, in 5 ears at 9%, the balance is A P r n nt,000 0.09 (5) $8,76.. Formula for compound interest Substitute for P, Use a calculator. and t. b. For monthl compounding, ou have n. So, in 5 ears at 9%, the balance is A P r n nt,000 0.09 (5) $8,788.7. c. For continuous compounding, the balance is A Pe rt,000e 0.09(5) $8,89.75. Now tr Eercise 59. Formula for compound interest Substitute for P, Use a calculator. and t. Formula for continuous compounding Substitute for P, r, and t. Use a calculator. In Eample 8, note that continuous compounding ields more than quarterl or monthl 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 per ear. r, n, r, n,
Section. Eample 9 Eponential Functions and Their Graphs Radioactive Deca The half-life of radioactive radium 6Ra 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 t 599 ears, then, is 5. a. What is the initial mass (when t 0)? b. How much of the initial mass is present after 500 ears? Graphical Algebraic t 599 5 5 0 599 a. 5 Use a graphing utilit to graph 5 t 599 Write original equation. a. Use the value feature or the zoom and trace features of the graphing utilit to determine that when 0, the value of is 5, as shown in Figure.. So, the initial mass is 5 grams. Substitute 0 for t. 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, as shown in Figure.. So, about 8.6 grams is present after 500 ears. Simplif. So, the initial mass is 5 grams. 5 t 599 b. 5 8.6 5. Write original equation. 0 0 500 599 Substitute 500 for t..56 Simplif. Use a calculator. 5000 0 So, about 8.6 grams is present after 500 ears. 0 FIGURE 0 5000 0. FIGURE. Now tr Eercise 7. CLASSROOM DISCUSSION Identifing Eponential Functions Which of the following functions generated the two tables below? Discuss how ou were able to decide. What do these functions have in common? Are an of them the same? If so, eplain wh. b. f!"! 8 c. f!"!! e. f5!"! 7 " f. f6!"! 8!" a. f!"!!"" d. f!"! " 7 " 0 0 g 7.5 8 9 5 h 6 8 Create two different eponential functions of the forms! a!b" and! c " d with -intercepts of!0, ".
Chapter Eponential and Logarithmic Functions. EXERCISES See www.calcchat.com for worked-out solutions to odd-numbered eercises. VOCABULARY: 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 the Propert to solve simple eponential equations.. 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. SKILLS AND APPLICATIONS In Eercises 7, evaluate the function at the indicated value of. Round our result to three decimal places. 7. 8. 9. 0... Function Value In Eercises 6, match the eponential function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) f 0.9. f. f 5 f 5 0 g 5000 f 00. 6 6 (0, ) (0, ) (b) (d). f. f 5. f 6. f.5 6 (0, ( 6 6 (0, ) In Eercises 7, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. 7. 8. 9. f 6 0. f 6. f. f In Eercises 8, use the graph of f to describe the transformation that ields the graph of g... 5. 6. 7. 8. In Eercises 9, use a graphing utilit to graph the eponential function. 9. 0... In Eercises 8, evaluate the function at the indicated value of. Round our result to three decimal places.. f f, f, f, f 0, f 7, f 0., Function h e g g g g 0 g 7 g 0. 5 Value. f e. 5. f e 5 0 6. f.5e 0 7. f 5000e 0.06 6 8. f 50e 0.05 0 f
Section. Eponential Functions and Their Graphs 5 In Eercises 9, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. 9. f e 0. f e. f e. f e 0.5. f e. f e 5 In Eercises 5 50, use a graphing utilit to graph the eponential function. 5..08 5 6..08 5 7. s t e 0.t 8. s t e 0.t 9. g e 50. h e In Eercises 5 58, use the One-to-One Propert to solve the equation for. 5. 7 5. 6 5. 5. 5 55. e e 56. e e 57. e e 58. e 6 e 5 COMPOUND INTEREST In Eercises 59 6, complete the table to determine the balance A for P dollars invested at rate r for t ears and compounded n times per ear. 59. P $500, r %, t 0 ears 60. P $500, r.5%, t 0 ears 6. P $500, r %, t 0 ears 6. P $000, r 6%, t 0 ears COMPOUND INTEREST In Eercises 6 66, complete the table to determine the balance A for $,000 invested at rate r for t ears, compounded continuousl. 6. r % 6. r 6% 5 n 65 Continuous A t 0 0 0 0 50 A 65. r 6.5% 66. r.5% 67. 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. 68. 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? 69. 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. 70. 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. 7. POPULATION GROWTH The projected populations of California for the ears 05 through 00 can be modeled b P.696e 0.0098t, 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? 7. 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 00. 7. RADIOACTIVE DECAY Let Q represent a mass of radioactive plutonium 9 Pu (in grams), whose halflife 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 50,000.
6 Chapter Eponential and Logarithmic Functions 7. RADIOACTIVE DECAY 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 (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 0,000. 75. 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. 76. 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. EXPLORATION TRUE OR FALSE? In Eercises 77 and 78, determine whether the statement is true or false. Justif our answer. 77. The line is an asmptote for the graph of f 0. 78. Q 0 t 575. e 7,80 99,990 THINK ABOUT IT In Eercises 79 8, use properties of eponents to determine which functions (if an) are the same. 79. f 80. f g 9 g 6 h 9 h 6 8. f 6 8. f e g g e h 6 8. Graph the functions given b and and use the graphs to solve each inequalit. (a) < (b) h e > 8. 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 85. GRAPHICAL ANALYSIS 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. 86. GRAPHICAL ANALYSIS 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? 87. GRAPHICAL ANALYSIS 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) 88. THINK ABOUT IT Which functions are eponential? (a) (b) (c) (d) 89. 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. 90. CAPSTONE 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 0 8 6 PROJECT: POPULATION PER SQUARE MILE To work an etended application analzing the population per square mile of the United States, visit this tet s website at academic.cengage.com. (Data Source: U.S. Census Bureau) d e f
Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn 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. Wh ou should learn it Logarithmic functions are often used to model scientific observations. For instance, in Eercise 97 on page 6, a logarithmic function is used to model human memor. Logarithmic Functions In Section.9, 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., 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 For > 0, a > 0, and a, log if and onl if a a. The function given b f log a The equations log a and Read as log base a of. is called the logarithmic function with base 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. Eample Evaluating Logarithms Use the definition of logarithmic function to evaluate each logarithm at the indicated value of. a. f log, b. f log, c. f log, d. f log 0, a. f log 5 because b. f log 0 because c. f log because. d. f 00 log 0 00 because 0 0 Now tr Eercise. 5. 0. 00 00.
8 Chapter 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. Eample Evaluating Common Logarithms on a Calculator Use a calculator to evaluate the function given b f log at each value of. a. 0 b. c..5 d. Function Value Graphing Calculator Kestrokes Displa a. f 0 log 0 LOG 0 ENTER f log b. LOG ENTER 0.77 c. f.5 log.5 LOG.5 ENTER 0.97900 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. Now tr Eercise 9. The following properties follow directl from the definition of the logarithmic function with base a. 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 Eample Using Properties of Logarithms a. Simplif: log b. Simplif: log 7 7 c. Simplif: 6 log 6 0 a. Using Propert, it follows that log 0. b. Using Propert, ou can conclude that log 7 7. c. Using the Inverse Propert (Propert ), it follows that 6 log 6 0 0. Now tr Eercise. You can use the One-to-One Propert (Propert ) to solve simple logarithmic equations, as shown in Eample.
Section. Logarithmic Functions and Their Graphs 9 Eample Using the One-to-One Propert a. log log Original equation One-to-One Propert b. c. log log log 6 log 0 6 0 6 ± Now tr Eercise 85. 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. Eample 5 Graphs of Eponential and Logarithmic Functions 0 8 6 6 8 0 FIGURE. f() = = g() = log In the same coordinate plane, sketch the graph of each function. a. f b. g log a. For f, construct a table of values. B plotting these points and connecting them with a smooth curve, ou obtain the graph shown in Figure.. 0 f 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.. Now tr Eercise 7. 8 5 Vertical asmptote: = 0 f() = log 5 6 7 8 9 0 FIGURE.5 Eample 6 Sketching the Graph of a Logarithmic Function Sketch the graph of the common logarithmic function f log. Identif the vertical asmptote. 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.5. The vertical asmptote is 0 ( -ais). 00 Without calculator With calculator 0 0 5 8 f log 0 0.0 0.699 0.90 Now tr Eercise.
0 Chapter Eponential and Logarithmic Functions The nature of the graph in Figure.5 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.6. FIGURE.6 (, 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 about the line a 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. Eample 7 Shifting Graphs of Logarithmic Functions You can use our understanding of transformations to identif vertical asmptotes of logarithmic functions. For instance, in Eample 7(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. 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.8. f() = log (, 0) (, ) (, 0) h() = + log f() = log You can review the techniques for shifting, reflecting, and stretching graphs in Section.7. g() = log( ) FIGURE.7 FIGURE.8 Now tr Eercise 5. (, 0)
Section. Logarithmic Functions and Their Graphs The Natural Logarithmic Function B looking back at the graph of the natural eponential function introduced on page 0 in Section., 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. (, e ) (0, ) f() = e (, e) (, 0) (, ) (e, ) = g() = f () = ln Reflection of graph of f e about the line FIGURE.9 e The Natural Logarithmic Function The function defined b f log e ln, > 0 is called the natural logarithmic function. 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.9. On most calculators, the natural logarithm is denoted b LN, as illustrated in Eample 8. Eample 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. WARNING / CAUTION 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. Function Value Graphing Calculator Kestrokes Displa a. f ln LN ENTER 0.697 b. f 0. ln 0. LN. ENTER.0978 c. f ln LN ENTER ERROR f ln d. LN ENTER 0.8876 Now tr Eercise 67. In Eample 8, be sure ou see that ln gives an error message on most calculators. (Some calculators ma displa a comple number.) This occurs because the domain of ln is the set of positive real numbers (see Figure.9). So, ln is undefined. The four properties of logarithms listed on page 8 are also valid for natural logarithms.
Chapter Eponential and Logarithmic Functions 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 Eample 9 Using Properties of Natural Logarithms Use the properties of natural logarithms to simplif each epression. a. ln ln b. e c. e ln 5 d. ln e a. ln e ln e Inverse Propert b. e ln 5 5 Inverse Propert c. ln 0 0 Propert d. ln e Propert Now tr Eercise 7. Eample 0 Finding the Domains of Logarithmic Functions Find the domain of each function. a. f ln b. g ln c. h ln a. Because ln is defined onl if > 0, it follows that the domain of f is,. The graph of f is shown in Figure.0. b. Because ln is defined onl if > 0, it follows that the domain of g is,. The graph of g is shown in Figure.. 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.. f() = ln( ) 5 FIGURE.0 FIGURE. g() = ln( ) h() = ln FIGURE. Now tr Eercise 75.
Section. Logarithmic Functions and Their Graphs Application Eample 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 Graphical a. The original average score was Use a graphing utilit to graph the model 75 6 ln. Then use the value or trace feature to approimate the following. f 0 75 6 ln 0 Substitute 0 for t. 75 6 ln Simplif. 75 6 0 Propert of natural logarithms 75. b. After months, the average score was f 75 6 ln 75 6 ln Substitute for t. Use a calculator. 68.. c. After 6 months, the average score was 75 6 ln 7 00 00 Simplif. 75 6.0986 f 6 75 6 ln 6 a. When 0, 75 (see Figure.). So, the original average score was 75. b. When, 68. (see Figure.). So, the average score after months was about 68.. c. When 6, 6. (see Figure.5). So, the average score after 6 months was about 6.. Substitute 6 for t. 0 0 FIGURE Use a calculator. 6.. 0. Simplif. 75 6.959 0 FIGURE. 00 0 0 FIGURE.5 Now tr Eercise 97. CLASSROOM DISCUSSION Analzing a Human Memor Model Use a graphing utilit to determine the time in months when the average score in Eample was 60. Eplain our method of solving the problem. Describe another wa that ou can use a graphing utilit to determine the answer.
Chapter Eponential and Logarithmic Functions. EXERCISES See www.calcchat.com for worked-out solutions to odd-numbered eercises. VOCABULARY: 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. SKILLS AND APPLICATIONS In Eercises 7, write the logarithmic equation in eponential form. For eample, the eponential form of log is 5 5 5 5. 7. log 6 8. log 7 9. log 0. log 9 8 000. log. log 6 8 5. log. log 8 6 8 In Eercises 5, write the eponential equation in logarithmic form. For eample, the logarithmic form of 8 is log 8. 5. 5 5 6. 69 7. 8 8. 9 7 9. 6 0. 6 6. 0. 0 0.00 In Eercises 8, evaluate the function at the indicated value of without using a calculator... 5. 6. 7. 8. Function f log f log 5 f log 8 f log g log a g log b Value 6 5 0 a b In Eercises 9, use a calculator to evaluate f log at the indicated value of. Round our result to three decimal places. 9. 7 0. 8 500..5. 96.75 In Eercises 6, use the properties of logarithms to simplif the epression.. log 7. log. 5. 6. 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) (c) log (b) (d) (e) (f ) 7 9 log 9 5
Section. Logarithmic Functions and Their Graphs 5 5. f log 6. 7. f log 8. 9. f log 50. f log f log f log 87. log log 5 88. 89. ln ln 90. 9. ln ln 9. log 5 log ln 7 ln 7 ln ln 6 In Eercises 5 58, write the logarithmic equation in eponential form. 5. 0.69... 5. In Eercises 59 66, write the eponential equation in logarithmic form. In Eercises 67 70, use a calculator to evaluate the function at the indicated value of. Round our result to three decimal places. 67. 68. 69. 70. ln Function f ln f ln g 8 ln g ln Value ln 5 57. ln 0 58. ln e 65. e 66. e 8. 0.7 0.05 In Eercises 7 7, evaluate g ln at the indicated value of without using a calculator. 7. e 5 7. e 7. e 5 6 7. e 5 0.96... 5. ln 7.95... 5. ln 0.0... 55. ln 50 5.5... 56. ln 08 6.988... 59. e 5.598... 60. e 7.890... 6. e.687... 6. e.956... 6. e 0.9 0.06... 6. e. 0.065... 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 5 77. 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 9 80. 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 5 6 86. log log 9 9. MONTHLY PAYMENT The model t 6.65 ln 750, 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. 9. 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 0.055. (a) Complete the table and interpret our results. (b) Sketch a graph of the function. 95. 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 0.55 0.98t ln t, t 6 where t represents the ear, with t corresponding to 00. (Source: Warren Communication News) (a) Complete the table. > 750 K 6 8 0 t t 5 6 C (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.
6 Chapter Eponential and Logarithmic Functions 96. 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. (b) Use a graphing utilit to graph the function. 97. HUMAN MEMORY 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? 98. SOUND INTENSITY The relationship between the number of decibels and the intensit of a sound I in watts per square meter is (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. (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. EXPLORATION TRUE OR FALSE? In Eercises 99 and 00, determine whether the statement is true or false. Justif our answer. 99. You can determine the graph of f log 6 b graphing g 6 and reflecting it about the -ais. 00. The graph of f log contains the point 7,. In Eercises 0 0, 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? 0. 0. 0. 0. r 0.005 0.00 0.05 0.00 0.05 0.00 t 0 log I f, f 5, f e, f 8, 0. 0 g log g log 5 g ln g log 8 05. THINK ABOUT IT Complete the table for f 0. 0 f Complete the table for f log. 00 f 0 0 00 Compare the two tables. What is the relationship between f 0 and f log? 06. GRAPHICAL ANALYSIS 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 07. (a) Complete the table for the function given b f ln. 5 0 0 0 0 6 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). 08. CAPSTONE The table of values was obtained b evaluating a function. Determine which of the statements ma be true and which must be false. 0 8 (a) (b) (c) is an eponential function of. is a logarithmic function of. is an eponential function of. (d) is a linear function of. 09. 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
Section. Properties of Logarithms 7. PROPERTIES OF LOGARITHMS What ou should learn 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. Wh ou should learn it Logarithmic functions can be used to model and solve real-life problems. For instance, in Eercises 87 90 on page, a logarithmic function is used to model the relationship between the number of decibels and the intensit of a sound. 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 change-of-base formula. 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. Eample Changing Bases Using Common Logarithms Dnamic Graphics/ Jupiter Images a. b. a. b. log 5 log Eample log 5 log log 5 log.979 0.6006.9 log log Use a calculator. Simplif. Now tr Eercise 7(a). ln 5 ln.888.869.9 ln ln log a log log a.0798.5850 0.00 Changing Bases Using Natural Logarithms log a ln ln a Use a calculator. Simplif..89.5850 0.695 Now tr Eercise 7(b).
8 Chapter 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. WARNING / CAUTION 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. 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.. Product Propert: Logarithm with Base a log a uv log a u log a v. Quotient Propert: log u a v log a u log a v Natural Logarithm ln uv ln u ln v ln u ln u ln v v. Power Propert: log ln u n a u n n log a u n ln u For proofs of the properties listed above, see Proofs in Mathematics on page 76. Eample Using Properties of Logarithms HISTORICAL NOTE The Granger Collection 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. a. Rewrite 6 as. Product Propert b. ln ln ln 7 Quotient Propert 7 Now tr Eercise 7. Rewrite 7 as. Power Propert 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 a. b. ln 6 ln Eample ln ln ln 7 ln ln ln ln log 5 5 log 5 5 log 5 5 ln e 6 ln e ln e6 e ln e ln e Now tr Eercise 9.
Section. Properties of Logarithms 9 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. Eample 5 Epanding Logarithmic Epressions Epand each logarithmic epression. a. log 5 b. ln 5 7 a. log 5 log 5 log log Product Propert log 5 log log Power Propert 5 5 b. ln ln Rewrite using rational eponent. 7 7 ln 5 ln 7 Quotient Propert ln 5 ln 7 Power Propert Now tr Eercise 5. 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. Eample 6 Condensing Logarithmic Epressions Condense each logarithmic epression. a. log log b. ln ln c. log log a. Power Propert log log log log log Product Propert b. ln ln ln ln Power Propert You can review rewriting radicals and rational eponents in Appendi A.. ln Quotient Propert c. Product Propert log log log Now tr Eercise 75. log log Power Propert Rewrite with a radical.
0 Chapter 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 the points are graphed and fall on a line, then ou can determine that the - and -values are related b the equation ln m ln where m is the slope of the line. Eample 7 Finding a Mathematical Model Period (in ears) 0 5 0 5 0 5 FIGURE.6 Planets Near the Sun Mercur Venus Earth Mars Saturn Jupiter 6 8 0 Mean distance (in astronomical units) 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, 0.87 0.7.000.5 5.0 9.57 Period, 0. 0.65.000.88.860 9.60 The points in the table above are plotted in Figure.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. This produces the following results. Venus Earth ln Mercur FIGURE.7 Mars Jupiter ln = 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). Choose an two points to determine the slope of the line. Using the two points 0., 0.6 and 0, 0, ou can determine that the slope of the line is m B the point-slope form, the equation of the line is Y where Y ln and X ln. You can therefore conclude that ln X, ln. Planet Mercur Venus Earth Mars Jupiter Saturn ln 0.99 0. 0.000 0..69.55 ln. 0.86 0.000 0.6.7.8 0.6 0 0. 0.5. Now tr Eercise 9.
Section. Properties of Logarithms. EXERCISES See www.calcchat.com for worked-out solutions to odd-numbered eercises. VOCABULARY 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. In Eercises 6, match the propert of logarithms with its name.. log a uv log a u log a v (a) Power Propert 5. ln u n n ln u (b) Quotient Propert u 6. log a (c) Product Propert v log a u log a v SKILLS AND APPLICATIONS In Eercises 7, rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. 7. log 5 6 8. log 7 9. log 5 0. log. log 0. log. log.6. log 7. In Eercises 5, evaluate the logarithm using the change-of-base formula. Round our result to three decimal places. 5. log 7 6. log 7 7. log 8. log 5 9. log 9 0. 0. log 0 0.5. log 5 50. log 0.05 In Eercises 8, use the properties of logarithms to rewrite and simplif the logarithmic epression.. log. log 8 5. log 5 6. 50 log 9 00 7. ln 5e 8. ln 6 6 e In Eercises 9, find the eact value of the logarithmic epression without using a calculator. (If this is not possible, state the reason.) log 6 9. log 9 0.. log 8. 6. log 6. log 8 5. log 6. log 7 log 5 5 7. ln e.5 8. ln e 9. ln 0. ln e e. ln e ln e 5. ln e 6 ln e 5. log 5 75 log 5. log log In Eercises 5 66, use the properties of logarithms to epand the epression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 5. ln 6. log 0z 7. log 8 8. 5 9. log 5 50. 5. ln z 5. ln t 5. ln z 5. log 55. ln z z, z > 56. a 6 57. log, a > 58. ln 9 59. 60. ln ln 6. ln 6. log z z 6. log 5 6. z log 0 log 6 z ln, > log 0 z 5 65. ln 66. ln
Chapter Eponential and Logarithmic Functions In Eercises 67 8, condense the epression to the logarithm of a single quantit. 67. ln ln 68. ln ln t 69. log z log 70. log 5 8 log 5 t 7. log log 7. log 7 z 7. log 5 7. log 6 75. log log 76. ln 8 5 ln z 77. 8. 8. In Eercises 85 and 86, compare the logarithmic quantities. If two are equal, eplain wh. 85. 86. log log log z 78. log log log z 79. ln ln ln 80. ln z ln z 5 ln z 5 8. ln ln ln 8. ln ln ln log 8 log 8 log 8 log log 6 log log log, log, log log log 7 70, log 7 5, SOUND INTENSITY In Eercises 87 90, use the following information. The relationship between the number of decibels and the intensit of a sound l in watts per square meter is given b 0 log I 0. log 7 0 87. 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. 88. Find the difference in loudness between an average office with an intensit of.6 0 7 watt per square meter and a broadcast studio with an intensit of.6 0 0 watt per square meter. 89. 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. 90. 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 9 9, find a logarithmic equation that relates and. Eplain the steps used to find the equation. 9. 9. 9. 9. 5 6.89.6..95.565 5 6.587.080.50.9.0 5 6.5.0.9.768.67.597 5 6 0.5.88 7.79 6 7.95.09 95. 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, 5 5 50 75 500 000 Galloping speed, 9.5 8.7 7.8 6. 5.9. 96. 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, 0.07 0.0 0.8 Length, Diameter, 0.0 5 0.8 6 0.8
Section. Properties of Logarithms 97. COMPARING MODELS A cup of water at an initial temperature of 78 C is placed in a room at a constant 00. CAPSTONE A classmate claims that the following temperature of C. The temperature of the water is are true. measured ever 5 minutes during a half-hour period. (a) ln u v ln u ln v ln uv The results are recorded as ordered pairs of the form t, T, where t is the time (in minutes) and T is the (b) ln u v ln u ln v ln u v temperature (in degrees Celsius). (c) ln u n n ln u ln u n 0, 78.0, 5, 66.0, 0, 57.5, 5, 5., Discuss how ou would demonstrate that these 0, 6., 5,., 0, 9.6 claims are not true. (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 5. 0.96 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? EXPLORATION 98. PROOF Prove that log b u v log b u log b v. 99. PROOF Prove that log b u n n log b u. TRUE OR FALSE? In Eercises 0 06, determine whether the statement is true or false given that f ln. Justif our answer. 0. 0. 0. 0. 05. If f u f v, then v u. 06. If f < 0, then 0 < <. In Eercises 07, use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utilit to graph the ratio. 07. 08. 09. 0... f 0 0 f a f a f, f f f, > f f f log f log f log f log f log.8 f log.. THINK ABOUT IT Consider the functions below. f ln, ln g, h ln ln ln a > 0, > 0 Which two functions should have identical graphs? Verif our answer b sketching the graphs of all three functions on the same set of coordinate aes.. GRAPHICAL ANALYSIS 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. 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).
Chapter Eponential and Logarithmic Functions. EXPONENTIAL AND LOGARITHMIC EQUATIONS What ou should learn 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. Wh ou should learn it Eponential and logarithmic equations are used to model and solve life science applications. For instance, in Eercise on page 5, an eponential function is used to model the number of trees per acre given the average diameter of the trees. 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 was used to solve simple eponential and logarithmic equations in Sections. 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 Eample Solving Simple Equations James Marshall/Corbis Original Rewritten Equation Equation 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 0 Inverse g. log log 8 Inverse Now tr Eercise 7. The strategies used in Eample are summarized as follows. Strategies for Solving Eponential 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.
Section. Eponential and Logarithmic Equations 5 Solving Eponential Equations Eample Solving Eponential Equations Solve each equation and approimate the result to three decimal places, if necessar. a. e e b. 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). a. e e Write original equation. 0 0 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. Write original equation. log log Divide each side b. Take log (base ) of each side. Inverse Propert Change-of-base formula The solution is log.807. Check this in the original equation. Now tr Eercise 9. 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. Eample 0 0 log ln ln.807 Solving an Eponential Equation Solve e 5 60 and approimate the result to three decimal places. Remember that the natural logarithmic function has a base of e. e 5 60 e 55 ln e ln 55 ln 55.007 Write original equation. Subtract 5 from each side. Take natural log of each side. Inverse Propert The solution is ln 55.007. Check this in the original equation. Now tr Eercise 55.