7Exponential and. Logarithmic Functions

Transcription

1 7Eponential and Logarithmic Functions A band of green light occasionall appears above the rising or setting sun. This phenomenon is known as a green flash because it lasts for a ver brief period of time. A green flash is caused b Earth s atmosphere refracting some of the sun s light. Wh does this phenomenon occur onl at sunrise or at sunset? You can use eponential and logarithmic functions to model man real-life situations. In Section 7.5, ou will learn about the tpes of data that are best represented b eponential growth models, Gaussian models, and logistic growth models. Pekka Parviainen/Polar Image 57

2 58 CHAPTER 7 Eponential and Logarithmic Functions Section 7. Eponential Functions and Their Graphs Recognize and evaluate eponential functions with base a. Graph eponential functions. Recognize and evaluate eponential functions with base e. Use eponential functions to model and solve real-life applications. Eponential Functions So far, this book has dealt onl 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. STUDY TIP For working with eponential functions, the following properties of eponents are useful. Properties of Eponents Let a and b be positive numbers.. a 0. a a a a. a a. a a 5. ab a b. a b 7. a a a b Definition of Eponential Function The eponential function 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 alread know how to evaluate a for integer and rational values of. For eample, ou know that and. However, to evaluate for an real number, ou need to interpret forms with irrational eponents. For the purposes of this book, it is sufficient to think of a where.5 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. EXAMPLE Graphs of a g() = In the same coordinate plane, sketch the graphs of f and g. f() = The table below lists some convenient 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. 0 8 Figure 7. The table feature of a graphing utilit could be used to epand the table.

3 SECTION 7. Eponential Functions and Their Graphs 59 G() = EXAMPLE Graphs of a In the same coordinate plane, sketch the graphs of F and G. 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. F() = 0 8 Figure 7. NOTE 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? In Eample, note that the functions given b F and G can be rewritten with positive eponents as the reciprocals of and. F and G 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

4 0 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 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() = g() = f() = (c) Reflection in the -ais Figure 7.5 (d) Reflection in the -ais 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.

5 SECTION 7. Eponential Functions and Their Graphs (, 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.. Be sure ou see that for the eponential function given b f e, e is the constant , whereas is the variable. (, e ) Figure 7. (, e ) THE NUMBER (0, ) e f() = 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. 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 EXAMPLE Graphing Natural Eponential Functions Sketch the graph of each natural eponential function. a. f e 0. b. 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 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)

6 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. P 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 A P r n nt. Amount (balance) with n compoundings per ear EXPLORATION Use the formula A P r n nt to calculate the amount in an account when P $000, r %, t 0 ears, and compounding is done () b the da, () b the hour, () b the minute, and () b the second. Use these results to present an argument that increasing the number of compoundings does not mean unlimited growth of the amount in the account. 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, 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, % should be epressed as 0.0. 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

7 SECTION 7. Eponential Functions and Their Graphs 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. 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. 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 c. For dail compoundings, ou have n 5. So, in 5 ears at 9%, the balance is A P r n nt, (5), (5), d. For continuous compounding, the balance is A Pe rt,000e 0.09(5) $8, $8,7.. $8, $8, 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. EXAMPLE Radioactive Deca Plutonium (in pounds) P Figure 7.8 P = 0( ) t/,0 (,0, 5) (00,000, 0.58) 50,000 00,000 Years of deca t In 98, a nuclear reactor accident occurred in Chernobl in what was then the Soviet Union. The eplosion spread highl toic radioactive chemicals, such as plutonium, over hundreds of square miles, and the government evacuated the cit and the surrounding area. To see wh the cit is now uninhabited, consider the model P 0 t,0. This model represents the amount of plutonium that remains (from an initial amount of 0 pounds) after t ears. Sketch the graph of this function over the interval from t 0 to t 00,000, where t 0 represents 98. How much of the 0 pounds will remain in the ear 00? How much of the 0 pounds will remain after 00,000 ears? The graph of this function is shown in Figure 7.8. Note from this graph that plutonium has a half-life of about,0 ears. That is, after,0 ears, half of the original amount will remain. After another,0 ears, one-quarter of the original amount will remain, and so on. In the ear 00 t, there will still be P 0, pounds of plutonium remaining. After 00,000 ears, there will still be P 0 00,000, pound of plutonium remaining.

8 CHAPTER 7 Eponential and Logarithmic Functions Eercises for Section 7. In Eercises 8, evaluate the epression. Round our result to three decimal places In Eercises 9, use the graph of f to describe the transformation that ields the graph of g In Eercises 7 0, match the eponential function with its graph. [The graphs are labeled (a) through (d).] (a) (b) (c) f, g f, g f, g 5 f 0, g 0 f 5, g 5 f 7, g 7 f 0., g 0. 5 f., g. 8 (d) 7. f 8. f 9. f 0. f In Eercises 8, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function b hand.. f. f. f. f 5. f. f 7. f 8. f In Eercises 9, use a graphing utilit to graph the eponential function. 9. g f 5. h. h 5. g 5. g f In Eercises 50, evaluate the epression. Round our result to three decimal places. e f e.. 5. e. e. 7. 0e e e e.5 In Eercises 5, use a graphing utilit to construct a table of values for the function. Then use a graphing utilit to graph the function. 5. f e 5. f e 5. f e 5. f e f e 5. f e s t e 0.t 58. s t e 0.t 59. g e 0. h e. f 000e 0.0. f 500e 0.0. f 50e f 50e f. f e 0.0 5e 0.0 Finance In Eercises 7 70, 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 5 Continuous A 7. P $500, r 8%, t 0 ears 8. P $000, r %, t 0 ears 9. P $500, r 8%, t 0 ears 70. P $000, r %, t 0 ears

9 SECTION 7. Eponential Functions and Their Graphs 5 Finance In Eercises 7 7, complete the table to determine the balance A for $,000 invested at rate r for t ears that is compounded continuousl. t A 7. r 8% 7. r % 7. r.5% 7. r 7.5% Writing About Concepts In Eercises 75 78, use properties of eponents to determine which functions (if an) are the same. 75. f 7. f g 9 g h 9 h 77. f 78. f 5 g g 5 h h Graph the functions given b and and use the graphs to solve the inequalities. (a) < (b) > 80. Graph the functions given b and and use the graphs to solve the inequalities. (a) < (b) > 8. Use a graphing utilit to graph e and each of the functions,,, and 5. Which function increases at the greatest rate as approaches? 8. Use the result of Eercise 8 to make a conjecture about the rate of growth of and n e, where n is a natural number and approaches. 8. Use the results of Eercises 8 and 8 to describe what is implied when it is stated that a quantit is growing eponentiall. 8. Which functions are eponential? (a) f (b) f (c) f (d) f 85. Finance On the da of a child s birth, a deposit of $5,000 is made in a trust fund that earns 8.75% interest that is compounded continuousl. Determine the balance in this account on the child s 5th birthda. 8. Finance An alumni of a college deposits $5000 in a trust fund that earns 7.5% interest that is compounded continuousl. It is specified that the balance will be given to the college in 50 ears. How much will the college receive? 87. Graphical Reasoning There are two options for investing $500. The first option earns 7% interest that is compounded annuall and the second option earns 7% simple interest. The figure shows the growth of each investment over a 0-ear period. (a) Identif which graph in the figure represents each tpe of investment. Eplain our reasoning. Investment (in dollars) (b) Verif our answer in part (a) b finding the equations that model the investment growth and then graph the models. 88. Depreciation After t ears, the value of a car is modeled b V t 0,000 t, where V is the value. (a) Graph the function, (b) determine the original value of the car, and (c) determine the value ears after it was purchased. 89. Inflation If the annual rate of inflation averages % over the net 0 ears, the approimate cost 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. If the price of an oil change for our car is presentl $.95, estimate the price 0 ears from now. 90. Economics The demand equation for a product is p Year e (a) Use a graphing utilit to graph the demand function for > 0 and p > 0. (b) Find the price p for a demand of 500 units. (c) Use the graph in part (a) to approimate the greatest price that will still ield a demand of at least 00 units.

10 CHAPTER 7 Eponential and Logarithmic Functions 9. Population Growth A certain tpe of bacterium increases according to the model P t 00e 0.97t, where t is the time in hours. Find (a) P 0, (b) P 5, and (c) P Population Growth The population of a town increases according to the model P t 500e 0.09t, where t is the time in ears, with t 0 corresponding to 990. Use the model to estimate the population in (a) 000 and (b) 00. Population P 9. Radioactive Deca Let Q (in grams) represent a mass of radioactive radium Ra, whose half-life is 599 ears. The quantit of radium present after t ears is given b Q 5 t 599. (a) Determine the initial quantit (when t 0). (b) Determine the quantit present after 000 ears. (c) Use a graphing utilit to graph the function over the interval t 0 to t Radioactive Deca Let Q (in grams) represent a mass of carbon C, whose half-life is 575 ears. The quantit of carbon present after t ears is given b 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, Data Analsis A meteorologist measures the atmospheric pressure P (in pascals) at altitude h (in kilometers). The data are shown in the table. A model for the data is given b P 07,8e 0.50h. = 500e 0.09t Year (0 990) h P 0,9 5,75,9, (a) Plot the data and the model on the same set of aes. (b) Create a table that compares the model with the sample data. t (c) Estimate the atmospheric pressure at a height of 8 kilometers. (d) Use the graph in part (a) to estimate the altitude at which the atmospheric pressure is,000 pascals. 9. Data Analsis To estimate the amount of defoliation caused b the gps moth during a given ear, a forester counts the number of egg masses on 0 of an acre (circle of radius 8. feet) in the fall. The percent of defoliation the net spring is shown in the table. (Source: USDA, Forest Service) A model for the data is given b 00 7e (a) Use a graphing utilit to plot the data and the model in the same viewing window. (b) Create a table that compares the model with the sample data. (c) Estimate the percent of defoliation if egg masses are counted on acre. 0 (d) Use the graph in part (a) to estimate the number of egg masses per 0 acre if ou observe that of a forest is defoliated the following spring. 97. Population The population of Las Vegas, Nevada P (in thousands) for selected ears from 970 to 000 can be modeled b P 5.5e 0.05t, where t is the time in ears, with t 0 corresponding to 970. (Source: U.S. Census Bureau) (a) Use a graphing utilit to graph the model. (b) Find the population for the ears 970, 980, 990, and 000. Eplain wh the data does not fit a linear model. (c) Use the graph from part (a) to estimate the ear when the population of Las Vegas is epected to eceed 750 thousand. 98. Vital Statistics The United States infant mortalit rate M (per 000 live births) for selected ears from 90 to 000 can be modeled b M 5.7e 0.09t, where t is the time (in ears), with t 0 corresponding to 90. (Source: U.S. National Center for Health Statistics) (a) Use a graphing utilit to graph the model. (b) Use the graph from part (a) to estimate when the mortalit rate was approimatel 0 per 000 live births. (c) Find the ear in which the mortalit rate decreased to 5 per 000 live births.

11 SECTION 7. Eponential Functions and Their Graphs 7 (d) Does the graph of the model have a t-intercept? Eplain our answer in the contet of the problem. 99. Population The table shows the African American population P (in millions) in the United States for selected ears from 850 to 000, where t is the time (in ears), with t 50 corresponding to 850. (Source: U.S. Census Bureau) t P t P t P (a) Use a graphing utilit to create a scatter plot of the data. (b) The data can be modeled b P.879e 0.05t. Use a graphing utilit to graph the model and the data in the same viewing window. (c) According to the model, when will the African American population in the United States surpass 0 million? 00. Population The table shows the population P (in millions) of people 5 ears old and older in the United States for selected ears from 950 to 000, where t is the time (in ears), with t 50 corresponding to 950. (Source: U.S. Census Bureau) t P (a) Use a graphing utilit to create a scatter plot of the data. (b) The data can be modeled b P.e 0.008t. Use a graphing utilit to graph the model and the data in the same viewing window. (c) According to the United States Census Bureau, the projected populations of people 5 ears old and older for the ears 00, 00, 00, 00, and 050 are 0., 5., 7.5, 80.05, and 8.7 million, respectivel. Does the model agree with these projections? Eplain our reasoning. 0. The death rate per one hundred thousand people in the United States from 970 to 00 due to heart disease can be modeled b D e 0.0t where t is the time (in ears) with t 0 corresponding to 970. (Source: U.S. National Center for Health Statistics) (a) Find the death rates in the ears 970, 980, 990, and 000. (b) Do ou think this model could be used to predict the death rate for the ears beond 00? Eplain our reasoning. 0. The table shows the number of cellular telephone subscribers C (in thousands) in the United States from 990 to 000, where t is the time (in ears), with t 0 corresponding to 990. (Source: Cellular Telecommunications Industr Association.) The data can be modeled b C,9.90,88.79e 0.0t. (a) Use a graphing utilit to graph the model and the data in the same viewing window. (b) According to this model, when will the number of cellular telephone subscribers eceed 50 million? True or False? In Eercises 0 0, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 0. The -ais is an asmptote for the graph of f The range of the eponential function given b f a is 0,. 05. e 7,80 99,990 e 0. e e 07. Use a graphing utilit to graph each function. Use the graph to find an asmptotes of the function. (a) t C 58,78 09,78 f 8 e 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 (b) g 8 e 0.5

12 8 CHAPTER 7 Eponential and Logarithmic Functions Section 7. Logarithmic Functions and Their Graphs Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions. Recognize and evaluate natural logarithmic functions. Use logarithmic functions to model and solve real-life applications. 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. Def inition 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 is called the logarithmic function with base a. The equations log a and a are equivalent. The first equation is in logarithmic form and the second is in eponential form. 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 a. f log 5 because 5. b. f log 0 because 0. c. f log because. d. f 00 log 0 00 because

13 SECTION 7. Logarithmic Functions and Their Graphs 9 The logarithmic function with base 0 is called the common logarithmic function. 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 0 at each value of. a. 0 b. c..5 d. Function Value Graphing Calculator Kestrokes Displa a. f 0 log 0 0 LOG 0 ENTER f log 0 b. LOG ENTER 0.77 c. f.5 log 0.5 LOG.5 ENTER d. f log 0 LOG ENTER ERROR Note that the calculator displas an error message (or a comple number) when ou tr to evaluate log 0. 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. EXPLORATION 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 0? 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 log 0. 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 log 0 0.

14 70 CHAPTER 7 Eponential and Logarithmic Functions 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, as indicated in the Eploration on the preceding page. EXAMPLE Graphs of Eponential and Logarithmic Functions 0 8 Figure 7.9 f() = = g() = log 8 0 In the same coordinate plane, sketch the graph of each function. a. f b. g log a. For f, construct a table of values. 0 f 8 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 given b Identif the -intercept and the vertical asmptote. f log 0. 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 log Vertical asmptote: = (, 0) f() = log 0 Figure 7.0

15 SECTION 7. Logarithmic Functions and Their Graphs 7 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 >. In Figure 7.0, ou would need to move out to 000 before the graph rose to. The basic characteristics of logarithmic graphs are summarized in Figure 7.. = log a (, 0) Figure 7. Graph of log a, a > Domain: 0, Range:, Intercept:, 0 Increasing -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 reviewed below to illustrate the inverse relation between the functions given b 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 b ± log a c. Notice how a horizontal shift of the graph results in a horizontal shift of the vertical asmptote. STUDY TIP You can use our understanding of transformations to identif vertical asmptotes of logarithmic functions. For instance, in Eample (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 asmptote of the graph of f. EXAMPLE Shifting Graphs of Logarithmic Functions The graph of each of the functions is similar to the graph of f log 0, as shown in Figure 7.. a. Because g log 0 f, the graph of g can be obtained b shifting the graph of f one unit to the right. b. Because h log 0 f, the graph of h can be obtained b shifting the graph of f two units up. f() = log 0 (, ) h() = + log 0 (, 0) (, 0) f() = log 0 g() = log 0 ( ) (, 0) (a) Figure 7. (b)

16 7 CHAPTER 7 Eponential and Logarithmic Functions 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. 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 9 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 a. ln Inverse Propert e ln e 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 e ln

17 SECTION 7. Logarithmic Functions and Their Graphs 7 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. Function Value Graphing Calculator Kestrokes Displa a. f ln LN ENTER 0.97 b. f 0. ln 0. LN. ENTER.0978 c. f ln LN ENTER ERROR d. LN ENTER f ln 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 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.

18 7 CHAPTER 7 Eponential and Logarithmic Functions Application EXAMPLE 0 Human Memor Model Students participating in a pschological 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 ln t, 0 t where t is the time (in months). The graph of f is shown in Figure 7.5. f(t) Average score f(t) = 75 ln(t + ) Time (in months) Figure 7.5 t 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 months? a. The original average score was f 0 75 ln 0 Substitute 0 for t. 75 ln Simplif Propert of natural logarithms 75. b. After months, the average score was f 75 ln Substitute for t. 75 ln Simplif Use a calculator. 8.. c. After months, the average score was f 75 ln 75 ln Substitute for t. Simplif. Use a calculator.

19 SECTION 7. Logarithmic Functions and Their Graphs 75 Eercises for Section 7. In Eercises 8, write the logarithmic equation in eponential form. For eample, the eponential form of log5 5 is log. log 8. log 7 9. log log. log log 8. log 8 7 (c) (d) In Eercises 9 8, write the eponential equation in logarithmic form. For eample, the logarithmic form of 8 is log a k b 8. u v w In Eercises 9 0, evaluate the epression without using a calculator. 9. log 0. log 8. log. log 7 9. log 7. log log log log 8 8. log 9 9. log a a 0. log b b In Eercises, use a calculator to evaluate the logarithm. Round to three decimal places.. log 0 5. log 0 5. log 0 5. log 0.5 In Eercises 5 0, use the graph of log to match the given function with its graph. [The graphs are labeled (a) through (f).] (a) (b) (e) 5. f log. f log 7. f log 8. f log 9. f log 0. f log In Eercises 50, find the domain, -intercept, and vertical asmptote of the logarithmic function and sketch its graph.. f log. g log. log. h log 5. f log. log 5 (f) log 0 5 log 0 9. f log f log 0 In Eercises 5 5, write the logarithmic equation in eponential form. 5. ln ln ln 0 5. ln e 55. ln e 5. ln e In Eercises 57, write the eponential equation in logarithmic form. 57. e e e 0 0. e e e.95...

20 7 CHAPTER 7 Eponential and Logarithmic Functions In Eercises 8, evaluate the epression without using a calculator.. ln e. ln e 5. ln e.. ln e ln ln e In Eercises 9 7, use a calculator to evaluate the logarithm. Round to three decimal places. 9. ln ln 7. ln ln ln 7. ln In Eercises 77 8, use the graph of ln to match the given function with its graph. [The graphs are labeled (a) through (f).] (a) (b) (c) (e) ln (d) 77. f ln 78. f ln 79. f ln 80. f ln 8. f ln 8. f ln (f) ln 5 e 5 5 In Eercises 8 90, find the domain, -intercept(s), and vertical asmptote(s) of the logarithmic function. Then use a graphing utilit to graph the function and verif our results. 8. f ln 8. h ln 85. g ln 8. f ln 87. f ln 88. f 89. f ln 90. f ln In Eercises 9 9, (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. 9. f ln 9. h ln ln 9. f 9. g ln Writing About Concepts g? In Eercises 95 98, describe the relationship between the graphs of f and g. What is the relationship between the functions f and 95. f 9. f 5 g log g log f e 98. f 0 g ln g log 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 00. The table of values was obtained b evaluating a function. Determine which of the statements ma be true and which must be false. 8 0 (a) is an eponential function of. (b) is a logarithmic function of. (c) is an eponential function of. (d) is a linear function of.

21 SECTION 7. Logarithmic Functions and Their Graphs 77 Writing About Concepts (continued) 0. Answer the following questions for the function given b f log 0. Do not use a calculator. (a) What is the domain of f? (b) What is f? (c) If is a real number between 000 and 0,000, in which interval will f be found? (d) In which interval will be found if f is negative? (e) If f is increased b unit, must have been increased b what factor? (f) If f n and f n, what is the ratio of to? 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 0 t, where t is the time in months. (a) What was the average score on the original eam t 0? (b) What was the average score after months? (c) What was the average score after 0 months? 0. Population Growth The population of a town will double in t 0 ln ln 7 ln 50 ears. Find t. 0. 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. 0 t r t (b) Use a reference source to decide which value of r best approimates the actual rate of growth for the world population. 05. Finance A principal P, invested at 9 % 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. K 8 0 t (b) Sketch a graph of the function. 0. Work The work (in foot-pounds) done in compressing a volume of 9 cubic feet at a pressure of 5 pounds per square inch to a volume of cubic feet is given b W 9,0 ln 9 ln. Find W. Ventilation In Eercises 07 and 08, use the logarithmic model 80. ln, , which approimates the minimum required ventilation rate in terms of the air space per child in a public school classroom. In the model, is the air space per child in cubic feet and is the ventilation rate in cubic feet per minute. 07. Use a graphing utilit to graph the function and approimate the required ventilation rate if there is 00 cubic feet of air space per child. 08. A classroom is designed for 0 students. The air conditioning sstem in the room has the capacit of moving 50 cubic feet of air per minute. (a) Determine the ventilation rate per child, assuming that the room is filled to capacit. (b) Use the graph in Eercise 07 to estimate the air space required per child. (c) Determine the minimum number of square feet of floor space required for the room if the ceiling height is 0 feet. 09. (a) Complete the table for the function given b f ln. 5 0 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).

22 78 CHAPTER 7 Eponential and Logarithmic Functions 0. Sound Intensit The relationship between the number of decibels and the intensit of a sound I in watts per square meter is given b (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 0 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 our reasoning. Monthl Pament In Eercises, use the model which approimates the length of a home mortgage of $50,000 at 8% 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). See figure. Length of mortgage (in ears) 0 log 0 t.5 ln 000, t I 0. > ,000 Monthl pament (in dollars). Use the model to approimate the length of a $50,000 mortgage at 8% if the monthl pament is $ Use the model to approimate the length of a $50,000 mortgage at 8% if the monthl pament is $5.8.. Approimate the total amount paid over the term of the mortgage in Eercise with a monthl pament of $00.5. What is the total interest charge?. Approimate the total amount paid over the term of the mortgage in Eercise with a monthl pament of $5.8. What is the total interest charge? 5. The median age in ears of the United States population from 99 to 00 can be modeled b.0 0. t ln t where t is the time (in ears), with t corresponding to 99. (Source: U.S. Census Bureau) (a) Use a graphing utilit to graph the model over the interval t. (b) Find the median age in 99 and in 00. (c) Do ou think the model would be accurate for ears beond 00? Eplain our reasoning. (d) Eplain wh the model cannot be used for ears prior to 99.. Skill Retention Model Participants in an industrial pscholog stud were taught a simple mechanical task and tested monthl on this mechanical task for a period of ear. The average scores for the class are given b the model f t 95 log 0 t, 0 t where t is the time (in months). (a) What was the average score on the original eam t 0? (b) What was the average score after months? (c) Use a graphing utilit to graph the model and approimate when the average score is 8. True or False? In Eercises 7 0, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 7. You can determine the graph of f log b graphing g and reflecting it about the -ais. 8. The graph of f log contains the point 7,. 9. The domain of the logarithmic function given b f log b c where b and c are positive constants is c b,. 0. If > 0, then e ln and ln e.

23 SECTION 7. Using Properties of Logarithms 79 Section 7. Using Properties of Logarithms Rewrite logarithmic functions with a different base. 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 applications. 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 logs and natural logs are the most frequentl used, ou ma occasionall need to evaluate logarithms to other bases. To do this, ou can use the following change-of -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 log a log b log b a Base 0 log a log 0 log 0 a Base e log a ln ln a One wa to look at the change-of-base formula is that logarithms to base a are simpl constant multiples of logarithms to base b. The constant multiplier is log b a. EXAMPLE Changing Bases Using Common Logarithms a. b. log 0 log 0 0 log log log 0 log log a log 0 log 0 a Use a calculator. Simplif. EXAMPLE Changing Bases Using Natural Logarithms a. log 0 b. log ln 0 ln ln ln log a b ln b ln a Use a calculator. Simplif

24 80 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 is not equal to log a log a. THEOREM 7. 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. log a uv log a u log a v. ln uv ln u ln v. log u a. v log a u log a v ln u ln u ln v v. 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 Write each logarithm in terms of ln and ln. a. ln b. ln 7 The Granger Collection JOHN NAPIER (550 7) 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. a. ln ln ln ln Rewrite as. Propert b. ln ln ln 7 7 ln ln ln ln Propert Rewrite 7 as. Propert EXAMPLE Using Properties of Logarithms Use the properties of logarithms to verif that log 0 00 log 0 00 log 0 00 log 0 00 log 0 00 log Rewrite 00 as 00. Propert Simplif.

25 SECTION 7. Using Properties of Logarithms 8 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. log 5 b. ln a. log 5 log 5 log log log 5 log log Propert Propert b. ln 5 7 ln ln 5 ln 7 Rewrite using a rational eponent. Propert ln 5 ln 7 Propert In Eample 5, the properties of logarithms were used to epand logarithmic epressions. In Eample, this procedure is reversed and the properties of logarithms are used to condense logarithmic epressions. EXAMPLE Condensing Logarithmic Epressions EXPLORATION Use a graphing utilit to graph the functions and ln ln ln in the same viewing window. Does the graphing utilit show the functions with the same domain? If so, should it? Eplain our reasoning. Condense each logarithmic epression. a. log 0 log 0 b. ln ln c. log log a. log 0 log 0 log 0 log 0 Propert log 0 Propert b. ln ln ln ln Propert ln Propert c. log log log Propert Propert log log Rewrite using a radical.

26 8 CHAPTER 7 Eponential and Logarithmic Functions Application One method of determining how the - and -values for a set of nonlinear data are related begins b taking the natural log of each of the - and -values. If these new points are graphed and fall on a straight 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 straight line. EXAMPLE 7 Finding a Mathematical Model Period (in ears) Mercur Venus Figure 7. Earth Mars Saturn Jupiter 8 0 Mean distance (in astronomical units) The table shows the mean distance 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 terms of ears. Find an equation that epresses as a function of. Planet Mercur Venus Earth Mars Jupiter Saturn Period Mean Distance The points in the table are plotted in Figure 7.. From this figure it is not clear how to find an equation that relates and. To solve this problem, take the natural log of each of the - and -values in the table. For instance, ln 0.. and ln Continuing this produces the following results. ln Planet Mercur Venus Earth Mars Jupiter Saturn ln ln Venus Earth Mars Jupiter Saturn ln = 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. Using properties of logarithms, ou can solve for as shown below. Mercur ln ln Propert One-to-One Propert Figure 7.7 The graph of this equation is shown in Figure 7.7.

27 SECTION 7. Using Properties of Logarithms 8 Eercises for Section 7. In Eercises 8, evaluate the logarithm using the change-of-base formula. Round our result to three decimal places.. log 7. log 7. log. log 5 5. log log log log 0.05 In Eercises 9, rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. 9. log 5 0. log. log 5. log. log. log 0 5. log.. log 7. In Eercises 7, use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utilit to sketch the graph of the function. 7. f log 8. f log 9. f log 0. f log. f log.8. f log. In Eercises, use the properties of logarithms to epand the epression as a sum, difference, and/or constant multiple of logarithms.. log 0 5. log 0 0z 5. log log ln z 0.. ln z.. ln a, a >. 5. ln z z, z >. ln ln ln 9. ln 0. ln z 5 log 0 log z ln t ln z. log b. log z b ln, > z In Eercises 0, condense the epression to the logarithm of a single quantit.. ln ln. ln ln t 5. log z log. log 5 8 log 5 t 7. log 8. log 9. log log 7 z ln ln ln 8 5 ln z ln ln ln ln ln z ln ln ln ln z ln z 5 ln z ln ln ln 58. ln ln ln 59. ln ln ln 0. ln ln ln In Eercises, condense the epression to the logarithm of a single quantit and simplif.. ln ln ln 9. ln ln ln. ln ln ln. ln ln ln In Eercises 5 and, compare the logarithmic quantities. If two are equal, eplain wh. 5.. log log, log, log log log 7 70, log 7 5, log 7 0 In Eercises 7 80, find the eact value of the logarithm without using a calculator. (If this is not possible, state the reason.) 7. log 9 8. log 9. log. 70. log log 9 7. log 7. log 5 75 log 5 7. log log 75. ln e ln e 5 7. ln e 77. log ln 79. ln e ln e

28 8 CHAPTER 7 Eponential and Logarithmic Functions In Eercises 8 8, use the properties of logarithms to rewrite and simplif the logarithmic epression. 8. log 8. log 8 8. log ln 5e 8. ln e In Eercises 87 9, use the given logarithms to approimate the logarithm. log b 0.89 log b log b log b 89. log b log b log 9. log b b b 5b 9. log 5 9. log b b b b 5 Writing About Concepts 98. Human Memor Model Students participating in a pschological eperiment attended several lectures 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 can be modeled b the memor model f t 90 5 log 0 t, log t log b In Eercises 95 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. 95. f log 0 g ln ln 0 9. f ln g log 0 log 0 e 97. 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. where t is the time in months. (a) What was the average score on the original eam t 0? (b) What was the average score after months? (c) What was the average score after months? (d) When did the average score decrease to 75? (e) Use the properties of logarithms to write the function in another form. (f) Sketch the graph of the function over the specified domain. 99. Sound Intensit The relationship between the number of decibels and the intensit of a sound I in watts per square meter is given b 0 log 0 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 watt per square meter. 00. Use a graphing utilit to graph each pair of functions in the same viewing window. Are the graphs identical? Eplain our reasoning. (a) f ln g ln (b) f ln g ln True or False? In Eercises 0 0, determine whether the statement is true or false given that f ln. If it is false, eplain wh or give an eample that shows it is false f 0 0 f a f a f, a > 0, > f f f, f f > 05. If f u f v, then v u. 0. If f < 0, then 0 < <. 07. Prove Propert of Theorem 7.: log a u v log a u log a v. 08. Prove Propert of Theorem 7.: log a u n n log a u. I 0.

29 SECTION 7. Eponential and Logarithmic Equations 85 Section 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 applications. 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 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 0 log Inverse The strategies used in Eample are summarized below. 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.

30 8 CHAPTER 7 Eponential and Logarithmic Functions Solving Eponential Equations EXAMPLE Solving Eponential Equations Solve each equation and approimate the result to three decimal places. a. e 7 b. c. e TECHNOLOGY When solving an eponential or logarithmic equation, remember that ou can check our solution graphicall b graphing the left and right sides separatel and using the intersect feature of our graphing utilit to determine the point of intersection. For instance, to check the solution of the equation in Eample (a), ou can graph and in the same viewing window, as shown in Figure 7.8. Using the intersect feature of our graphing utilit, ou can determine that the graphs intersect when.77, which confirms the solution found in Eample (a). 00 e = 7 7 a. e 7 ln e ln 7 ln 7.77 Write original equation. Take natural log of each side. Inverse Propert Use a calculator. The solution is ln Check this in the original equation. b. log log log ln ln.807 Write original equation. Divide each side b. Take log (base ) of each side. Inverse Propert Change-of-Base Formula Use a calculator. The solution is log.807. Check this in the original equation. c. e ln e ln ln ln 0. Write original equation. Take natural log of each side. Inverse Propert Subtract from each side. Use a calculator. The solution is ln 0.. Check this in the original equation. 0 0 Figure 7.8 = e 5 In Eample (a), the eact solution is ln 7 and the approimate solution is.77. 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 Solve e 5 0 and approimate the result to three decimal places. 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. e 5 0 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.

31 SECTION 7. Eponential and Logarithmic Equations 87 EXAMPLE Solving an Eponential Equation Solve t 5 and approimate the result to three decimal places. 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 Inverse Propert t 5 log 5 Add 5 to each side. t 5 log 5 Divide each side b. t.7 The solution is t 5 log 5 Use a calculator..7. 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 solutions. EXAMPLE 5 Solving an Eponential Equation of Quadratic Tpe Solve e e 0. 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 e 0 Set nd factor equal to 0. 0 The solutions are ln and 0. Check these in the original equation. Figure 7.9 To check the result of Eample 5 graphicall, ou can use a graphing utilit to graph e e. In Figure 7.9, note that the graph has two -intercepts: one at ln 0.9 and one at 0.

32 88 CHAPTER 7 Eponential and Logarithmic Functions Solving Logarithmic Equations To solve a logarithmic equation such as ln Logarithmic form write the equation in eponential form as follows. e ln e e Eponentiate each side. Eponential form This procedure is called eponentiating each side of an equation. EXAMPLE Solving a Logarithmic Equation Solve for. a. ln b. log 5 log 7 a. ln e ln e e Write original equation. Eponentiate each side. Inverse Propert The solution is e. Check this in the original equation. b. Write original equation. log 5 log One-to-One Propert 8 Add and to each side. Divide each side b. The solution is. Check this in the original equation. EXAMPLE 7 Solving a Logarithmic Equation Solve 5 ln and approimate the result to three decimal places. 5 ln ln Write original equation. Subtract 5 from each side. = ln e ln e e Divide each side b. Eponentiate each side. Inverse Propert 0.07 Use a calculator. (e /, ) The solution is e Check this in the original equation. 0 0 = 5 + ln To check the result of Eample 7 graphicall, ou can use a graphing utilit to graph 5 ln and Figure 7.0 in the same viewing window, as shown in Figure 7.0.

33 SECTION 7. Eponential and Logarithmic Equations 89 EXAMPLE 8 Solving a Logarithmic Equation Solve log 5. log 5 log 5 5 log Write original equation. Divide each side b. Eponentiate each side (base 5). Inverse Propert = ( 5, ) 5 Divide each side b. The solution is 5. Check this in the original equation. = log 5 0 To check the result of Eample 8 graphicall, graph the functions given b log 5 and Figure 7. in the same viewing window. The two graphs should intersect when as shown in Figure and, 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 s Solve log 0 5 log 0. log 0 5 log 0 Write original equation. log 0 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 Set nd factor equal to 0. The solutions appear to be 5 and. However, when ou check these in the original equation or use a graphical check, ou can see that 5 is the onl solution. STUDY TIP In Eample 9, the domain of log 0 5 is > 0 and the domain of log 0 is >, so the domain of the original equation is >. Because the domain is all real numbers greater than, the solution is etraneous.

34 90 CHAPTER 7 Eponential and Logarithmic Functions Applications EXAMPLE 0 Doubling an Investment You have deposited $500 in an account that pas.75% interest that is compounded continuousl. How long will it take our mone to double? Using the formula for continuous compounding, ou can find that the balance in the account is A Pe rt 500e 0.075t. To find the time required for the balance to double, let A 000 and solve the resulting equation for t. 500e 0.075t 000 Substitute 000 for A. e 0.075t Divide each side b 500. ln e 0.075t 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 Account balance (in dollars) (0.7, 000) (0, 500) 8 0 Time (in ears) t Figure 7. EXPLORATION 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. a. 7% annual interest rate that is compounded annuall b. 7% annual interest rate that is compounded continuousl c. 7% annual interest rate that is compounded quarterl d. 7.5% annual interest rate that is compounded quarterl Which savings plan has the greatest effective ield? Which savings plan will have the highest balance after 5 ears?

35 SECTION 7. Eponential and Logarithmic Equations 9 EXAMPLE Average Salar for Secondar Teachers The average salaries for secondar teachers (in thousands of dollars) from 980 to 00 can be modeled b the equation..9 ln t, 0 t where t 0 represents 980 (see Figure 7.). During which ear did the average salar for secondar teachers reach.5 times its 980 level of $.5 thousand? (Source: National Education Association) Average salar (in thousands of dollars) Year (0 980) 5 t Figure ln t Write original equation...9 ln t.5 Let ln t 8.55 Add. to each side. ln t.55 Divide each side b.9. e ln t e.55 Eponentiate each side. t e.55 Inverse Propert t 9 Use a calculator. The solution is t 9 ears. Because t 0 represents 980, it follows that the average salar for secondar teachers reached.5 times its 980 level in 999. Eercises for Section In Eercises, determine whether each -value is a solution (or approimate solution) of the equation.. 7. (a) 5 (a) (b) (b). e 75. (a) e 5 (a).5 log 5 8 (b) ln 5 (b) 0.58 (c).9 (c) ln 8 ln 5 5. log. ln.8 (a).5 (a) e.8 (b) (b) 5.70 (c) (c) ln.8 In Eercises 7 0, solve for

36 9 CHAPTER 7 Eponential and Logarithmic Functions ln ln 0 0. ln ln 5 0. e. e. ln. ln 7 5. log. log 5 7. log log log 0. log In Eercises, approimate the point of intersection of the graphs of f and g. Then solve the equation f g analticall.. f. f 7 g 8 g 9 8. f log. f ln g g 0 8 f g f g In Eercises 5, appl the Inverse Properties of logarithms and eponents to simplif the epression. 5. log 0 0. log 7. 8 log 8 8. log 9. ln e 7 0. ln e. e ln 5. eln. ln e. 8 eln In Eercises 5 5, solve the eponential equation with the natural base e analticall. Approimate the result to three decimal places. 5. e 0. e e 5 8. e 8 8 g 8 f f g 8 9. e 50. e e e e e e 5e e e In Eercises 57, solve the eponential equation with base a analticall. Approimate the result to three decimal places t 0.0. t In Eercises 5 7, use a graphing utilit to graph the function. Approimate its zero to three decimal places. 5. g e 5. f e 5 7. f e 9 8. g 8e 9. g t e 0.09t 70. f e h t e 0.5t 8 7. f e.7 9 In Eercises 7 8, solve the eponential equation. Approimate the result to three decimal places t t e e t t 0 In Eercises 8 00, solve the natural logarithmic equation analticall. Approimate the result to three decimal places. 8. ln 8. ln 85. ln. 8. ln 87. ln ln ln 90. ln ln ln ln ln ln ln ln ln ln ln

37 SECTION 7. Eponential and Logarithmic Equations 9 9. ln ln 97. ln ln 98. ln ln 99. ln 5 ln ln 00. ln ln ln In Eercises 0 0, solve the logarithmic equation analticall. Approimate the result to three decimal places. 0. log 0 z 0. log 0 0. log log log 0 log 0 log 0 0. log log log 07. log log 08. log log log 0 8 log 0 0. log 0 log In Eercises, use a graphing utilit to approimate (to three decimal places) the point of intersection of the graphs e.. 0 ln ln Writing About Concepts 5. Write the steps needed to solve an eponential equation.. Write the steps needed to solve a logarithmic equation. 7. You invest P dollars at an annual interest rate of r that is 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. 8. Write a paragraph eplaining whether the time required for an investment to double depends on the size of the investment. Writing About Concepts (continued) In Eercises 9 and 0, (a) find the time required for a $000 investment to double at interest rate r that is compounded continuousl. (b) Is the time required for the investment to quadruple twice as long as the time for it to double? (c) Support our answer from part (b) b suggesting a reason. (d) Verif our answer from part (b) analticall. 9. r r 0. Finance In Eercises and, find the time required for a $000 investment to triple at interest rate r, that is compounded continuousl.. r r 0.. Economics The demand equation for a product is p 7000 e Find the demand for a price of (a) p $50 and (b) p $00.. Economics The demand equation for a product is p 5000 Find the demand for a price of (a) p $00 and (b) p $ Forest Yield The ield V (in millions of cubic feet per acre) for a forest at age t ears is given b V.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 age of a forest with a ield of. million cubic feet per acre.. Trees per Acre The number of trees per acre N of a certain species is approimated b the model N , e where is the average diameter of the trees feet above the ground. Use the model to approimate the average diameter of the trees in a test plot when N.

38 9 CHAPTER 7 Eponential and Logarithmic Functions 7. Average Heights The percent of American males m between the ages of 8 and who are no more than inches tall is given b m The percent of American females f between the ages of 8 and who are no more than inches tall is given b f where is the height in inches. Center for Health Statistics) (Source: U.S. National (a) Use the graph to determine an horizontal asmptotes of the functions. What do the mean? (b) What is the median height of each se? Percent of population e e Human Learning Model 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 function. Interpret the meaning of the upper asmptote in the contet of this problem. (c) After how man trials will 0% of the responses be correct? 9. Data Analsis An object at a temperature of 0 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 these data is T 0 7 h. f() m() Height (in inches) h 0 5 T (a) The graph of this model is shown in the figure. 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 T = 0[ + 7( h )] Time (in hours) 0. Data Analsis The personal consumption medical care ependitures E (in billions of dollars) for selected ears from 90 to 000 are shown in the table. t E A model for these data is E 0.9e 0.0t, where t is the time in ears, with t 0 corresponding to 90. (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 005, 00, and 05. (c) Analticall 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.. The average monthl high temperature (in degrees Fahrenheit) in Mrtle Beach, South Carolina can be approimated b the model.5.5e t 7..9, t where t represents the month, with t corresponding to Januar. During which months of the ear is the high temperature likel to rise above 80? h

39 SECTION 7. Eponential and Logarithmic Equations 95. Finance The table shows the number N (in thousands) of banks in the United States from 995 to 00. The data can be modeled b the logarithmic function N ln t, where t represents the ear, with t 5 corresponding to 995. (Source: Federal Deposit Insurance Corp.) Year Number, N Year Number, N (a) Use the model to determine during which ear the number of banks reached 0,000. (b) Use a graphing utilit to graph the model. (c) Use the graph from part (b) to verif our answer in part (a).. The total world population P (in billions) b decade is shown in the table. (Source: U.S. Census Bureau International Database) Year Population Year Population A model for these data is P 8.7.5e 0.080t.0005 where t is the time in ears, with t 0 representing 950. (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 world population in 00, 00, and 00. (c) Analticall find the ear, according to the model, when the world population reaches 8 billion. (d) Graphicall determine the larger horizontal asmptote, and interpret its meaning in the contet of the model. (e) Do ou believe that future world population can be predicted using the given model? Eplain our reasoning.. Debt The gross federal debt D (in billions of dollars) for the United States for selected ears from 90 to 000 is shown in the table. (Source: Office of Management and Budget) t D t D A model for these data is D 9.7,95.e 0.98t 0.5 where t is the time in ears, with t 0 corresponding to 90. (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 gross federal debt for 005, 00, and 05. (c) Analticall find the ear, according to the model, when the gross federal debt reached trillion dollars. (d) Graphicall determine the larger horizontal asmptote, and interpret its meaning in the contet of the problem. (e) Analticall find the ear, according to the model, when the gross federal debt eceeds 0 trillion dollars, if ever. Eplain our result with respect to our answer in part (d). (f) Do ou believe that the future gross federal debt can be predicted using the given model? Eplain our reasoning. True or False? In Eercises 5 8, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 5. 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. 7. The logarithm of the difference of two numbers is equal to the difference of the logarithms of the numbers. 8. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers.

40 9 CHAPTER 7 Eponential and Logarithmic Functions Section 7.5 STUDY TIP In this section, ou will investigate these models graphicall and algebraicall. In Section 8., ou will stud some of these models from a calculus viewpoint. 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. Introduction The five most common tpes of mathematical models involving eponential functions and logarithmic functions are shown below.. 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 0 The graphs of the basic forms of these functions are shown in Figure 7.. = e = e = e = + e = + ln = + log 0 Figure 7. You can often gain quite a bit of insight into a situation modeled b an eponential or logarithmic function b simpl identifing and interpreting the function s asmptotes. Use the graphs in Figure 7. to identif the asmptotes of each function.

41 SECTION 7.5 Eponential and Logarithmic Models 97 Eponential Growth and Deca As shown on the preceding page, the models for eponential growth and deca var onl in the sign of the real number b. Eponential Growth: ae b, a > 0, b > 0 Eponential Deca: ae b, a > 0, b < 0 EXAMPLE Population Increase Population (in millions) P Year (5 995) Figure 7.5 P Population (in millions) Year (5 995) Figure 7. t t The world population (in millions) from 995 through 00 is shown in the table. The scatter plot of the data is shown in Figure 7.5. (Source: U.S. Census Bureau) Year Population An eponential growth model that approimates these data is P 50e 0.09t, 5 t where P is the population (in millions) and t 5 represents 995. Compare the values given b the model with the data given b the U.S. Census Bureau. According to this model, when will the world population reach.8 billion? The following table compares the two sets of population figures. The graph of the model with the data is shown in Figure 7.. Year Population Model To find when the world population will reach.8 billion, let P 800 in the model and solve for t. 50e 0.09t P Write original model. 50e 0.09t 800 Substitute 800 for P. e 0.09t.7 Divide each side b 50. ln e 0.09t ln.7 Take natural log of each side. 0.09t 0.98 Inverse Propert t 8.7 Divide each side b According to the model, the world population will reach.8 billion in 008. TECHNOLOGY Some graphing utilities have curve-fitting capabilities that can be used to find models that represent data. If ou have such a graphing utilit, tr using it to find a model for the data given in Eample. How does our model compare with the model given in Eample?

42 98 CHAPTER 7 Eponential and Logarithmic Functions 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 Population Figure 7.7 Ratio 0 (0 ) 0 (, 00) 5 Time (in das) R Figure 7.8 = e 0.59t t = 0 (, 00) (5, 5) t = 5700 t = 9, ,000 Time (in ears) t t 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? 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 aebt produces 00 ae b and 00 ae b. To solve for b, solve for a in the first equation. 00 ae b Then substitute the result into the second equation. 00 ae b e b eb 00 eb 00 ln b ln b Solve for a in the first equation. Write second equation. Substitute 00 for a. 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 as shown in Figure 7.7. This implies that, after 5 das, the population will be e flies. 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). R 8 0e t 00 e ln a 00 e b 00. Carbon dating model The graph of R is shown in Figure 7.8. Note that R decreases as t increases. e b

43 SECTION 7.5 Eponential and Logarithmic Models 99 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 00, the Scholastic Aptitude Test (SAT) math scores for college-bound seniors roughl followed a normal distribution 0.005e 5 5,58, where is the SAT score for mathematics. (Source: College Board) a. Sketch the graph of this function. b. From the graph, estimate the average SAT math score. a. The graph of the function is shown in Figure 7.9. Distribution = 5 50% of population Figure Score b. From the graph, ou can see that the average mathematics score for college-bound seniors in 00 was 5.

44 500 CHAPTER 7 Eponential and Logarithmic Functions 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 7.0. 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 placed under less favorable conditions that inhibit growth. A logistic growth curve is also called a sigmoidal curve. EXAMPLE Spread of a Virus Figure Students infected (0., 000) 000 (5, 5) Time (in das) Figure 7. t 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 e 0.8t, t 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? a. After 5 das, the number of students infected is e e 5. b. Classes will be canceled when the number infected is e 0.8t.5 e 0.8t e 0.8t ln e 0.8t ln t ln t.5 ln t 0. So, after 0 das, at least 0% of the students will be infected, and classes will be canceled. The graph of the function is shown in Figure 7..

45 SECTION 7.5 Eponential and Logarithmic Models 50 Logarithmic Models EXAMPLE 5 Magnitudes of Earthquakes AP/Wide World Photo Twent seconds of a 7. magnitude earthquake in Kobe, Japan, on Januar 7, 995, left damage approaching $0 billion. On the Richter scale, the magnitude R of an earthquake of intensit I is R log 0 I I 0 where I 0 is the minimum intensit used for comparison. Find the intensities per unit of area for the following earthquakes. (Intensit is a measure of the wave energ of an earthquake.) a. Toko and Yokohama, Japan, in 9: R 8. b. Kobe, Japan, in 995: R 7. c. El Salvador in 00: R 7.7 a. Because I 0 and R 8., ou have 8. log I log 0 I I ,5,000. b. For R 7., ou have 7. log I log 0 I I ,89,000. c. For R 7.7, ou have 7.7 log I 0 Substitute for Eponentiate each side. Inverse Propert Substitute for Eponentiate each side. Inverse Propert Substitute for I 0 I 0 I 0 and 8. for R. and 7. for R. and 7.7 for R log 0 I Eponentiate each side. I Inverse Propert 50,9,000. Note that an increase of. units on the Richter scale (from 7. to 8.) represents an increase in intensit b a factor of 99,5,000 5,89,000. In other words, the earthquake in 9 had an intensit about times greater than that of the 995 quake.