Exponential and Logarithmic Functions

Transcription

1 Eponential and Logarithmic Functions. Eponential Functions and Their Graphs. Logarithmic Functions and Their Graphs. Properties of Logarithms. Eponential and Logarithmic Equations.5 Eponential and Logarithmic Models Carbon dating is a method used to determine the ages of archeological artifacts up to 50,000 ears old. For eample, archeologists are using carbon dating to determine the ages of the great pramids of Egpt. Slvain Grandadam/Gett Images SELECTED APPLICATIONS Eponential and logarithmic functions have man real-life applications. The applications listed below represent a small sample of the applications in this chapter. Computer Virus, Eercise 5, page 7 Data Analsis: Meteorolog, Eercise 70, page 8 Sound Intensit, Eercise 90, page 8 Galloping Speeds of Animals, Eercise 85, page Average Heights, Eercise 5, page 55 Carbon Dating, Eercise, page IQ Scores, Eercise 7, page Forensics, Eercise, page 8 Compound Interest, Eercise 5, page 7 7

2 8 Chapter Eponential and Logarithmic Functions. Eponential Functions and Their Graphs What ou should learn Recognize and evaluate eponential functions with base a. Graph eponential functions and use the One-to-One Propert. Recognize,evaluate, and graph eponential functions with base e. Use eponential functions to model and solve real-life problems. Wh ou should learn it Eponential functions can be used to model and solve real-life problems. For instance, in Eercise 70 on page 8, an eponential function is used to model the atmospheric pressure at different altitudes. Eponential Functions So far, this tet has dealt mainl with algebraic functions, which include polnomial functions and rational functions. In this chapter, ou will stud two tpes of nonalgebraic functions eponential functions and logarithmic functions. These functions are eamples of transcendental functions. Definition of Eponential Function The eponential function f with base a is denoted b f a where a > 0, a, The base a is ecluded because it ields f. This is a constant function, not an eponential function. You have evaluated a for integer and rational values of. For eample, ou know that and. However, to evaluate for an real number, ou need to interpret forms with irrational eponents. For the purposes of this tet, it is sufficient to think of a and is an real number. (where.5) as the number that has the successivel closer approimations a., a., a., a., a.,.... Eample Evaluating Eponential Functions Comstock Images/Alam Use a calculator to evaluate each function at the indicated value of. Function Value a. b. c. f f f 0.. Function Value Graphing Calculator Kestrokes Displa a. f... ENTER 0.9 b. f ENTER 0.7 c. f 0.. ENTER > > > Now tr Eercise. When evaluating eponential functions with a calculator, remember to enclose fractional eponents in parentheses. Because the calculator follows the order of operations, parentheses are crucial in order to obtain the correct result.

3 Section. Eponential Functions and Their Graphs 9 Eploration Note that an eponential function f a is a constant raised to a variable power, whereas a power function g n is a variable raised to a constant power. 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., Graphs of Eponential Functions The graphs of all eponential functions have similar characteristics, as shown in Eamples,, and 5. Eample Graphs of a In the same coordinate plane, sketch the graph of each function. a. f b. g The table below lists some values for each function, and Figure. 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. g() = 0 8 f() = FIGURE. G() = 0 8 F() = FIGURE. Now tr Eercise. The table in Eample was evaluated b hand. You could, of course, use a graphing utilit to construct tables with even more values. Eample Graphs of a In the same coordinate plane, sketch the graph of each function. a. F b. G The table below lists some values for each function, and Figure. 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. Now tr Eercise. In Eample, note that b using one of the properties of eponents, the functions F and G can be rewritten with positive eponents. F and G

4 0 Chapter Eponential and Logarithmic Functions Notice that the range of an eponential function is 0,, which means that a > 0 for all values of. 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. and. 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. and.. = a (0, ) Graph of a, a > Domain:, Range: 0, Intercept: 0, Increasing -ais is a horizontal asmptote a 0 as Continuous FIGURE. (0, ) = a Graph of a, a > Domain:, Range: 0, Intercept: 0, Decreasing -ais is a horizontal asmptote a 0 as Continuous FIGURE. From Figures. and., ou can see that the graph of an eponential function is alwas increasing or alwas decreasing. As a result, the graphs pass the Horizontal Line Test, and therefore the functions are one-to-one functions. You can use the following One-to-One Propert to solve simple eponential equations. For a > 0 and a, a a if and onl if. One-to-One Propert Eample Using the One-to-One Propert a. 9 Original equation 9 One-to-One Propert Solve for. b. 8 Now tr Eercise 5.

5 Section. Eponential Functions and Their Graphs In the following eample, notice how the graph of a sketch the graphs of functions of the form f b ± a c. can be used to Eample 5 Transformations of Graphs of Eponential Functions Each of the following graphs is a transformation of the graph of f. a. Because g f, the graph of g can be obtained b shifting the graph of f one unit to the left, as shown in Figure.5. b. Because h f, the graph of h can be obtained b shifting the graph of f downward two units, as shown in Figure.. c. Because k f, the graph of k can be obtained b reflecting the graph of f in the -ais, as shown in Figure.7. d. Because j f, the graph of j can be obtained b reflecting the graph of f in the -ais, as shown in Figure.8. g() = + f() = f() = h() = FIGURE.5 Horizontal shift FIGURE. Vertical shift f() = k() = j() = f() = FIGURE.7 Reflection in -ais FIGURE.8 Reflection in -ais Now tr Eercise 7. Notice that the transformations in Figures.5,.7, and.8 keep the -ais as a horizontal asmptote, but the transformation in Figure. ields a new horizontal asmptote of. Also, be sure to note how the -intercept is affected b each transformation.

6 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 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 , whereas is the variable. Eploration 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. Eample Evaluating the Natural Eponential Function 8 7 f() = e 0. 5 FIGURE g() = e 0.58 FIGURE. Use a calculator to evaluate the function given b f e at each indicated value of. a. b. c. 0.5 d. 0. Function Value Graphing Calculator Kestrokes Displa a. f e e ENTER 0.55 b. f e e ENTER c. f 0.5 e 0.5 e 0.5 ENTER.805 d. f 0. e 0. e 0. ENTER Eample 7 Now tr Eercise 7. 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 Figures.0 and.. Note that the graph in Figure.0 is increasing, whereas the graph in Figure. is decreasing. 0 f g Now tr Eercise 5.

7 Section. Eponential Functions and Their Graphs Eploration 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 (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 amount in the account? Eplain. 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 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 a ear. If the interest is added to the principal at the end of the ear, the new balance is P P P Pr P r. This pattern of multipling the previous principal b r is then repeated each successive ear, as shown below. Year Balance After Each Compounding 0 P P P P r 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 , , , ,000, ,000, m m 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.

8 Chapter Eponential and Logarithmic Functions Be sure ou see that the annual interest rate must be written in decimal form. For instance, % should be written 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 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, (5) $8,7.. 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, (5) Formula for compound interest Substitute for P, and t. $8, Use a calculator. c. For continuous compounding, the balance is A Pe rt Formula for continuous compounding,000e 0.09(5) Substitute for P, r, and t. $8, Use a calculator. Now tr Eercise 5. 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 a ear. r, n, r, n,

9 Section. Eponential Functions and Their Graphs 5 Eample 9 Radioactive Deca Plutonium (in pounds) P FIGURE. Radioactive Deca ( t/,00 P = 0 ( (,00, 5) (00,000, 0.5) 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,00 which represents the amount of plutonium P 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.. Note from this graph that plutonium has a half-life of about,00 ears. That is, after,00 ears, half of the original amount will remain. After another,00 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 00,000,00 of plutonium remaining pound Now tr Eercise 7. W RITING ABOUT MATHEMATICS 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. a. f ( ) b. c. f 8 f ( ) d. f e. f f. f g h 8 Create two different eponential functions of the forms with -intercepts of 0,. a b and c d

10 Chapter Eponential and Logarithmic Functions. Eercises VOCABULARY CHECK: Fill in the blanks.. Polnomials and rational functions are eamples of functions.. Eponential and logarithmic functions are eamples of nonalgebraic functions, also called functions.. The eponential function given b f e is called the function, and the base e is called the base.. 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. 5. To find the amount A in an account after t ears with principal P and an annual interest rate r compounded continuousl, ou can use the formula. In Eercises, evaluate the function at the indicated value of. Round our result to three decimal places. Function Value... f. 5. f. f 5. f g f 00. In Eercises 7 0, match the eponential function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) In Eercises, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function.... f. f 5. f. f In Eercises 7, use the graph of transformation that ields the graph of f f, g f, g f, g 5 f 0, g 0 f 7, g 7 f 0., g 0. 5 f f g. to describe the (c) (d) 7. f 8. f 9. f 0. f In Eercises, use a graphing utilit to graph the eponential function In Eercises 7, evaluate the function at the indicated value of. Round our result to three decimal places. Function Value 7. h e 8. f e. 9. f e f.5e 0. f 5000e 0.0. f 50e

11 Section. Eponential Functions and Their Graphs 7 In Eercises 8, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function.. f e. f e 5. f e. f e f e 8. f e 5 In Eercises 9, use a graphing utilit to graph the eponential function s t e 0.t. s t e 0.t. g e. h e In Eercise 5 5, use the One-to-One Propert to solve the equation for e e 50. e e 5. e e 5. e e 5 Compound Interest In Eercises 5 5, complete the table to determine the balance A for P dollars invested at rate r for t ears and compounded n times per ear. 5. P $500, r.5%, t 0 ears 5. P $000, r %, t 0 ears 55. P $500, r %, t 0 ears 5. P $000, r %, t 0 ears 5 n 5 Continuous A Compound Interest In Eercises 57 0, complete the table to determine the balance A for $,000 invested at rate r for t ears, compounded continuousl. t A 57. r % 58. r % 59. r.5% 0. r.5%. Trust Fund On the da of a child s birth, a deposit of $5,000 is made in a trust fund that pas 8.75% interest, compounded continuousl. Determine the balance in this account on the child s 5th birthda.. 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?. 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.. Demand The demand equation for a product is given b p 5000 where p is the price and is the number of units. (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. 5. Computer Virus The number V of computers infected b a computer virus increases according to the model V t 00e.05t, where t is the time in hours. Find (a) V, (b) V.5, and (c) V.. Population The population P (in millions) of Russia from 99 to 00 can be approimated b the model P 5.e 0.009t, where t represents the ear, with t corresponding to 99. (Source: Census Bureau, International Data Base) (a) According to the model, is the population of Russia increasing or decreasing? Eplain. (b) Find the population of Russia in 998 and 000. (c) Use the model to predict the population of Russia in Radioactive Deca Let Q represent a mass of radioactive radium Ra (in grams), whose half-life is 599 ears. The quantit of radium present after t ears is Q 5 t 599. e 0.00 (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 represent a mass of carbon C (in grams), whose half-life is 575 ears. The quantit of carbon present after t ears is Q 0 t 575. (a) Determine the initial quantit (when t 0). (b) Determine the quantit present after 000 ears. (c) Sketch the graph of this function over the interval t 0 to t 0,000.

12 8 Chapter Eponential and Logarithmic Functions 9. Data Analsis: Biolog 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 Egg masses, 00 7e Model It (a) Use a graphing utilit to create a scatter plot of the data and graph 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 Percent of defoliation, (d) You observe that of a forest is defoliated the following spring. Use the graph in part (a) to estimate the number of egg masses per acre. 70. Data Analsis: Meteorolog A meteorologist measures the atmospheric pressure P (in pascals) at altitude h (in kilometers). The data are shown in the table. Altitude, h Pressure, P 0 0,9 5 5,75 0,9 5,57 0 5,09 A model for the data is given b P 07,8e 0.50h. (a) Sketch a scatter plot of the data and graph the model on the same set of aes. (b) Estimate the atmospheric pressure at a height of 8 kilometers. 0 Snthesis True or False? In Eercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. The line is an asmptote for the graph of f e 7,80 99,990. Think About It In Eercises 7 7, use properties of eponents to determine which functions (if an) are the same. 7. f 7. f g 9 g h 9 h 75. f 7. f e g g e h 77. Graph the functions given b and and use the graphs to solve each inequalit. (a) < (b) > 78. 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 79. Graphical Analsis Use a graphing utilit to graph f 0.5 and in the same viewing window. What is the relationship between f and g as increases and decreases without bound? 80. Think About It Which functions are eponential? (a) (b) (c) (d) Skills Review In Eercises 8 and 8, solve for. h e g e In Eercises 8 and 8, sketch the graph of the function. 8. f 8. f Make a Decision To work an etended application analzing the population per square mile of the United States, visit this tet s website at college.hmco.com. (Data Source: U.S. Census Bureau)

13 Section. Logarithmic Functions and Their Graphs 9. Logarithmic Functions and Their Graphs 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 89 on page 8, 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 Read as log base a of. is called the logarithmic function with base a. Ariel Skelle/Corbis The equations log a and 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, Remember that a logarithm is an eponent. So, to evaluate the logarithmic epression log a, ou need to ask the question, To what power must a be raised to obtain? 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 0 0 Now tr Eercise

14 0 Chapter Eponential and Logarithmic Functions Eploration Complete the table for f 0. 0 f Complete the table for f log f 00 Compare the two tables. What is the relationship between f 0 and f log? 0 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 b. f log LOG ENTER 0.77 c. f.5 log.5 LOG.5 ENTER d. f log LOG ENTER ERROR Note that the calculator displas an error message (or a comple number) when ou tr to evaluate log. The reason for this is that there is no real number power to which 0 can be raised to obtain. Now tr Eercise. 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: log 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 log 0 0. Now tr Eercise 7. You can use the One-to-One Propert (Propert ) to solve simple logarithmic equations, as shown in Eample.

15 Section. Logarithmic Functions and Their Graphs Eample Using the One-to-One Propert a. log log Original equation One-to-One Propert b. log log c. log log 0 0 ± Now tr Eercise 79. 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 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 FIGURE. 5 FIGURE. Vertical asmptote: = 0 f() = log 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.. Eample Now tr Eercise. 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.. The vertical asmptote is 0 ( -ais). Without calculator Now tr Eercise 7. With calculator f log

16 Chapter Eponential and Logarithmic Functions The nature of the graph in Figure. 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.5. FIGURE.5 (, 0) = log a Graph of log a, a > Domain: 0, Range:, -intercept:, 0 Increasing One-to-one, therefore has an inverse function -ais is a vertical asmptote log as 0 a. Continuous Reflection of graph of a about the line The 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.. b. Because h log f, the graph of h can be obtained b shifting the graph of f two units upward, as shown in Figure.7. f() = log (, 0) (, ) (, 0) h() = + log f() = log g() = log( ) (, 0) FIGURE. FIGURE.7 Now tr Eercise 9.

17 Section. Logarithmic Functions and Their Graphs The Natural Logarithmic Function B looking back at the graph of the natural eponential function introduced in Section. on page 88, 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) (e, ) = (, 0) ( e, ) g() = f () = ln Reflection of graph of f e about the line FIGURE.8 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.8. On most calculators, the natural logarithm is denoted b LN, as illustrated in Eample 8. Eample 8 Evaluating the Natural Logarithmic Function 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. 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 f ln d. LN ENTER Now tr Eercise. 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.8). So, ln is undefined. The four properties of logarithms listed on page 0 are also valid for natural logarithms.

18 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 5. 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.9. b. Because ln is defined onl if > 0, it follows that the domain of g is,. The graph of g is shown in Figure.0. 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 g() = ln( ) h() = ln FIGURE.9 FIGURE.0 Now tr Eercise 9. FIGURE.

19 ft () Memor Model Application Section. Logarithmic Functions and Their Graphs 5 Average score f(t) = 75 ln(t + ) Time (in months) FIGURE. t 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 ln t, 0 t where t is the time in months. The graph of f is shown in Figure.. 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 Substitute for t. 75 ln 7 Simplif Use a calculator... Now tr Eercise 89. W RITING ABOUT MATHEMATICS Analzing a Human Memor Model Use a graphing utilit to determine the time in months when the average score in Eample was 0. Eplain our method of solving the problem. Describe another wa that ou can use a graphing utilit to determine the answer.

20 Chapter Eponential and Logarithmic Functions. Eercises VOCABULARY CHECK: 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 Propert of logarithms and eponentials states that log a a and. 5. The One-to-One Propert of natural logarithms states that if ln ln, then. In Eercises 8, write the logarithmic equation in eponential form. For eample, the eponential form of log is log 8. log log 5. log 5. log 8 7. log 8. log 8 In Eercises 9, write the eponential equation in logarithmic form. For eample, the logarithmic form of 8 is log In Eercises 7, evaluate the function at the indicated value of without using a calculator. Function 7. f log 8. f log 9. f log 7 0. f log. g log a. g log b Value 0 a b 000 In Eercises, use a calculator to evaluate f log at the indicated value of. Round our result to three decimal places In Eercises 7 0, use the properties of logarithms to simplif the epression. 7. log 8. log log 9 5 In Eercises 8, 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 7. log 8. log In Eercises 9, 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 5 (b) (d)

21 Section. Logarithmic Functions and Their Graphs 7 (e) 9. f log 0. f log. f log. f log. f log. f log In Eercises 5 5, write the logarithmic equation in eponential form ln ln ln ln ln ln 0 5. ln e In Eercises 5 0, write the eponential equation in logarithmic form. 5. e e e e e e e 0. e In Eercises, use a calculator to evaluate the function at the indicated value of. Round our result to three decimal places. Function. f ln. f ln. g ln. g ln (f) Value In Eercises 5 8, evaluate g ln at the indicated value of without using a calculator. 5. e. e 7. e 8. e 5 In Eercises 9 7, find the domain, -intercept, and vertical asmptote of the logarithmic function and sketch its graph. 9. f ln 70. h ln 7. g ln 7. f ln In Eercises 7 78, use a graphing utilit to graph the function. Be sure to use an appropriate viewing window. 7. f log 7. f log 75. f ln 7. f ln 77. f ln 78. f ln In Eercises 79 8, use the One-to-One Propert to solve the equation for. 79. log log 80. log log 9 8. log log 5 8. log 5 log 8. ln ln 8. ln ln 85. ln ln 8. ln ln 87. Monthl Pament The model Length of mortgage (in ears) 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) t Model It t.5 ln 000, > 000,000,000,000 8,000 0,000 Monthl pament (in dollars) (a) Use the model to approimate the lengths of a $50,000 mortgage at 8% when the monthl pament is $00.5 and when the monthl pament is $5.8. (b) Approimate the total amounts paid over the term of the mortgage with a monthl pament of $00.5 and with a monthl pament of $5.8. (c) Approimate the total interest charges for a monthl pament of $00.5 and for a monthl pament of $5.8. (d) What is the vertical asmptote for the model? Interpret its meaning in the contet of the problem.

22 8 Chapter Eponential and Logarithmic Functions 88. Compound Interest 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. (b) Sketch a graph of the function. 89. Human Memor Model Students in a mathematics class were given an eam and then retested monthl with an equivalent eam. The average scores for the class are given b the human memor model f t 80 7 log t, 0 t where t is the time in months. (a) Use a graphing utilit to graph the model over the specified domain. (b) What was the average score on the original eam t 0? (c) What was the average score after months? (d) What was the average score after 0 months? 90. Sound Intensit The relationship between the number of decibels and the intensit of a sound I in watts per square meter is 0 log I (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. Snthesis K 8 0 t 0. True or False? In Eercises 9 and 9, determine whether the statement is true or false. Justif our answer. 9. You can determine the graph of f log b graphing g and reflecting it about the -ais. 9. The graph of f log contains the point 7,. In Eercises 9 9, sketch the graph 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? f, f 5, f e, f 0, g log g log 5 g ln g log 97. Graphical Analsis Use a graphing utilit to graph f and g in the same viewing window and determine which is increasing at the greater rate as approaches. What can ou conclude about the rate of growth of the natural logarithmic function? (a) (b) f ln, f ln, g g 98. (a) Complete the table for the function given b f ln. (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). 99. Think About It The table of values was obtained b evaluating a function. Determine which of the statements ma be true and which must be false. (a) is an eponential function of. (b) is a logarithmic function of. (c) is an eponential function of. (d) is a linear function of. 00. Writing Eplain wh log a is defined onl for 0 < a < and a >. In Eercises 0 and 0, (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 h ln f ln Skills Review In Eercises 0 08, evaluate the function for f and g. 0. f g 0. f g 05. fg f f g f g g f 0

23 Section. Properties of Logarithms 9. 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 8 8 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 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. 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 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. Eample log a log log a log a ln ln a Changing Bases Using Common Logarithms AP Photo/Stephen Chernin a. b. log 5 log 5 log log a log log a.979 Use a calculator Simplif. log.0798 log.5850 log 0.00 Now tr Eercise (a). Eample Changing Bases Using Natural Logarithms a. b. log 5 ln 5 ln log a ln ln a.888 Use a calculator Simplif. ln.89 log.5850 ln 0.95 Now tr Eercise (b).

24 0 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. 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. Logarithm with Base a Natural Logarithm. Product Propert: log a uv log a u log a v ln uv ln u ln v. Quotient Propert: log a u v log a u log a v ln u ln u ln v v. Power Propert: log ln u n a u n n log a u n ln u For proofs of the properties listed above, see Proofs in Mathematics on page 78. Eample Using Properties of Logarithms The Granger Collection Historical Note 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 b. ln 7 a. ln ln ln ln Rewrite as. Product Propert b. ln ln ln 7 7 ln ln ln ln Quotient Propert Rewrite 7 as. Power Propert Now tr Eercise 7. Eample Using Properties of Logarithms Find the eact value of each epression without using a calculator. a. log b. ln e ln e 5 5 a. b. log 5 5 log 5 5 log 5 5 ln e ln e ln e e ln e ln e Now tr Eercise.

25 Rewriting Logarithmic Epressions Section. Properties of Logarithms 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 Eploration Use a graphing utilit to graph the functions given b 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. Epand each logarithmic epression. a. log 5 b. ln a. log 5 log 5 log log Product Propert b. ln 5 7 log 5 log log ln ln 5 ln 7 ln 5 ln 7 Now tr Eercise 7. Power Propert Rewrite using rational eponent. Quotient Propert Power 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. Eample Condensing Logarithmic Epressions Condense each logarithmic epression. a. log log b. ln ln c. log log a. log log log log log Power Propert Product Propert b. ln ln ln ln Power Propert ln Quotient Propert c. log log log Product Propert log log Power Propert Rewrite with a radical. Now tr Eercise 9.

26 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 Planets Near the Sun 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 ears. Find an equation that relates and. Period (in ears) FIGURE. Mercur Venus Earth Jupiter Mars Saturn 8 0 Mean distance (in astronomical units) Planet Mean Period, distance, Mercur Venus Earth Mars.5.88 Jupiter Saturn The points in the table above are plotted in Figure.. 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. Earth Venus ln Mercur FIGURE. 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.). Choose an two points to determine the slope of the line. Using the two points 0., 0. 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 ln Now tr Eercise 85.

27 Section. Properties of Logarithms. Eercises VOCABULARY CHECK: In Eercises and, 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. In Eercises 5, match the propert of logarithms with its name.. log a uv log a u log a v (a) Power Propert. ln u n n ln u (b) Quotient Propert u 5. log a (c) Product Propert v log a u log a v In Eercises 8, rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms.. log 5. log. log 5. log 5. log 0. log 7. log. 8. log 7. In Eercises 9, evaluate the logarithm using the change-of-base formula. Round our result to three decimal places. 9. log 7 0. log 7. log. log 5. log log log log 0.05 In Eercises 7, use the properties of logarithms to rewrite and simplif the logarithmic epression. 7. log 8. log log 5 0. log 50. ln 5e. ln e In Eercises 8, find the eact value of the logarithmic epression without using a calculator. (If this is not possible, state the reason.) 00. log log 8. log 7. log. 8. log log 9 0. log. ln e.5 log 5 5. ln e. ln e. ln e 5. ln e ln e 5. ln e ln e 5 7. log 5 75 log 5 8. log log In Eercises 9 0, use the properties of logarithms to epand the epression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 9. log 5 0. log 0z. log. log 8 0. log 5. log 5 z 5. ln z. ln t 7. ln z 8. log 9. ln z z, z > 50. a 5. log, a > 5. ln ln ln 55. ln 5. log z 5 z log z ln, > log 0 z ln 0. ln

28 Chapter Eponential and Logarithmic Functions In Eercises 78, condense the epression to the logarithm of a single quantit.. ln ln. ln ln t. log z log. log 5 8 log 5 t 5. log. log 7 z 7. log 5 8. log 9. ln ln 70. ln 8 5 ln z 7. log log log z 7. log log log z 7. ln ln ln 7. ln z ln z 5 ln z ln ln ln 7. ln ln ln 77. log 8 log 8 log log log log In Eercises 79 and 80, compare the logarithmic quantities. If two are equal, eplain wh. 79. log log, log, log log 80. log 7 70, log 7 5, Sound Intensit In Eercises 8 8, use the following information. The relationship between the number of decibels and the intensit of a sound I in watts per square meter is given b 0 log I 0. log Use the properties of logarithms to write the formula in simpler form, and determine the number of decibels of a sound with an intensit of 0 watt per square meter. 8. Find the difference in loudness between an average office with an intensit of. 0 7 watt per square meter and a broadcast studio with an intensit of. 0 5 watt per square meter. 8. 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? Model It 8. Human Memor Model Students participating in a pscholog 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 human memor model f t 90 5 log t, 0 t where t is the time in months. (a) Use the properties of logarithms to write the function in another form. (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 months? (e) Use a graphing utilit to graph the function over the specified domain. (f) Use the graph in part (e) to determine when the average score will decrease to 75. (g) Verif our answer to part (f) numericall. 85. Galloping Speeds of Animals Four-legged animals run with two different tpes of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times, whereas an animal that is galloping has all four feet off the ground at some point in its stride. The number of strides per minute at which an animal breaks from a trot to a gallop depends on the weight of the animal. Use the table to find a logarithmic equation that relates an animal s weight (in pounds) and its lowest galloping speed (in strides per minute). Weight, Galloping Speed,

29 Section. Properties of Logarithms 5 8. Comparing Models A cup of water at an initial temperature of 78 C is placed in a room at a constant temperature of C. The temperature of the water is measured ever 5 minutes during a half-hour period. The results are recorded as ordered pairs of the form t, T, where t is the time (in minutes) and T is the temperature (in degrees Celsius). 0, 78.0, 5,.0, 0, 57.5, 5, 5., 0,., 5,., 0, 9. (a) The graph of the model for the data should be asmptotic with the graph of the temperature of the room. Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utilit to plot the data points t, T and t, T. (b) An eponential model for the data t, T is given b T t. Solve for T and graph the model. Compare the result with the plot of the original data. (c) Take the natural logarithms of the revised temperatures. Use a graphing utilit to plot the points t, ln T and observe that the points appear to be linear. Use the regression feature of the graphing utilit to fit a line to these data. This resulting line has the form ln T at b. Use the properties of the logarithms to solve for T. 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) Write a short paragraph eplaining wh the transformations of the data were necessar to obtain each model. Wh did taking the logarithms of the temperatures lead to a linear scatter plot? Wh did taking the reciprocals of the temperature lead to a linear scatter plot? Snthesis True or False? In Eercises 87 9, determine whether the statement is true or false given that f ln. Justif our answer f 0 0 f a f a f, a > 0, > f f f, f f > 9. If f u f v, then v u. 9. If f < 0, then 0 < <. 9. Proof Prove that log u b v log b u log b v. 9. Proof Prove that log b u n n log b u. In Eercises 95 00, use the change -of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utilit to graph both functions in the same viewing window to verif that the functions are equivalent. 95. f log 9. f log 97. f log 98. f log 99. f log f log. 0. Think About It Consider the functions below. f ln ln, g, h ln ln ln 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. 0. Eploration For how man integers between and 0 can the natural logarithms be approimated given that ln 0.9, ln.098, and ln 5.09? Approimate these logarithms (do not use a calculator). Skills Review In Eercises 0 0, simplif the epression In Eercises 07 0, solve the equation

30 Chapter Eponential and Logarithmic Functions. Eponential 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 55, a logarithmic 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 Eample a log a log a a Solving Simple Equations Original Rewritten Equation Equation Propert James Marshall/Corbis 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 Inverse Now tr Eercise. 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.

31 Section. Eponential and Logarithmic Equations 7 Solving Eponential Equations Eample Solving Eponential Equations Solve each equation and approimate the result to three decimal places if necessar. a. e e b. a. e e Write original equation. One-to-One Propert Write in general form. Factor. Set st factor equal to 0. Set nd factor equal to 0. The solutions are and. Check these in the original equation. b. log log log ln.807 ln Write original equation. 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 5. 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 Solving an Eponential Equation Remember that the natural logarithmic function has a base of e. Solve e 5 0 and approimate the result to three decimal places. e 5 0 Write original equation. e 55 Subtract 5 from each side. ln e ln 55 Take natural log of each side. ln Inverse Propert The solution is ln Check this in the original equation. Now tr Eercise 5.

32 8 Chapter Eponential and Logarithmic Functions Eample Solving an Eponential Equation Solve t 5 and approimate the result to the nearest thousandth. 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. Remember that to evaluate a logarithm such as log 7.5, ou need to use the change-of-base formula. log 7.5 ln 7.5 ln.8 The solution is t 5 log 5 t 5 log 7.5 t 5 log 7.5 t.7 t 5 log Now tr Eercise 5. Inverse Propert Add 5 to each side. Divide each side b. Use a calculator. 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. Eample 5 Solving an Eponential Equation of Quadratic Tpe Solve e e 0. Algebraic 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 0.9 and 0. Check these in the original equation. Now tr Eercise 7. Graphical Use a graphing utilit to graph e e. Use the zero or root feature or the zoom and trace features of the graphing utilit to approimate the values of for which 0. In Figure.5, ou can see that the zeros occur at 0 and at 0.9. So, the solutions are 0 and 0.9. = e e + FIGURE.5

33 Section. Eponential and Logarithmic Equations 9 Solving Logarithmic Equations To solve a logarithmic equation, ou can write it in eponential form. ln e ln e e Logarithmic form Eponentiate each side. Eponential form This procedure is called eponentiating each side of an equation. Eample Solving Logarithmic Equations a. ln Original equation Remember to check our solutions in the original equation when solving equations to verif that the answer is correct and to make sure that the answer lies in the domain of the original equation. e ln e e Eponentiate each side. Inverse Propert b. log 5 log Original equation One-to-One Propert Add and to each side. Divide each side b. c. log log 5 log Original equation log 5 log Quotient Propert of Logarithms One-to-One Propert Cross multipl. Isolate. Divide each side b 7. Now tr Eercise 77. Eample 7 Solving a Logarithmic Equation Solve 5 ln and approimate the result to three decimal places. 5 ln Write original equation. ln Subtract 5 from each side. ln Divide each side b. e ln e e 0.07 Eponentiate each side. Inverse Propert Use a calculator. Now tr Eercise 85.

34 50 Chapter Eponential and Logarithmic Functions Eample 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 Divide each side b. Notice in Eample 9 that the logarithmic part of the equation is condensed into a single logarithm before eponentiating each side of the equation. The solution is 5. Check this in the original equation. Now tr Eercise 87. 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. Eample 9 Checking for Etraneous s Solve log 5 log. Algebraic Graphical log 5 log log 5 0 log Write original equation. Product Propert of Logarithms Eponentiate each side (base 0). Inverse Propert Write in general form. 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, ou can see that 5 is the onl solution. Now tr Eercise 99. Use a graphing utilit to graph log 5 log and in the same viewing window. From the graph shown in Figure., it appears that the graphs intersect at one point. Use the intersect feature or the zoom and trace features to determine that the graphs intersect at approimatel 5,. So, the solution is 5. Verif that 5 is an eact solution algebraicall. 0 5 = log 5 + log( ) FIGURE. = 9 In Eample 9, the domain of log 5 is > 0 and the domain of log is >, so the domain of the original equation is >. Because the domain is all real numbers greater than, the solution is etraneous. The graph in Figure. verifies this concept.

35 C WASHINGTON SERIES 99 C Applications Section. Eponential and Logarithmic Equations 5 Eample 0 Doubling an Investment Eploration 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, compounded annuall b. 7% annual interest rate, compounded continuousl c. 7% annual interest rate, compounded quarterl d. 7.5% annual interest rate, compounded quarterl Which savings plan has the greatest effective ield? Which savings plan will have the highest balance after 5 ears? You have deposited $500 in an account that pas.75% interest, 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 A 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 e 0.075t ln e 0.075t ln 0.075t ln t ln t 0.7 Let A 000. Divide each side b 500. Take natural log of each side. Inverse Propert Divide each side b Use a calculator. The balance in the account will double after approimatel 0.7 ears. This result is demonstrated graphicall in Figure.7. Account balance (in dollars) A FIGURE.7 THE UNITED STATES OF AMERICA (0, 500) Now tr Eercise 07. A Doubling an Investment WASHINGTON, D.C. (0.7, 000) A = 500e 0.075t 8 Time (in ears) 0 t In Eample 0, an approimate answer of 0.7 ears is given. Within the contet of the problem, the eact solution, ln ears, does not make sense as an answer.

36 5 Chapter Eponential and Logarithmic Functions Number of species Endangered Animal Species Year (0 990) FIGURE.8 t Eample Endangered Animals The number of endangered animal species in the United States from 990 to 00 can be modeled b 9 ln t, where t represents the ear, with t 0 corresponding to 990 (see Figure.8). During which ear did the number of endangered animal species reach 57? (Source: U.S. Fish and Wildlife Service) 9 ln t 9 ln t 57 ln t 7 0 t Write original equation. Substitute 57 for. Add 9 to each side. ln t 7 Divide each side b. e ln t e 7 Eponentiate each side. t e 7 Inverse Propert t 8 Use a calculator. The solution is t 8. Because t 0 represents 990, it follows that the number of endangered animals reached 57 in 998. Now tr Eercise. W RITING ABOUT MATHEMATICS Comparing Mathematical Models The table shows the U.S. Postal Service rates for sending an epress mail package for selected ears from 985 through 00, where 5 represents 985. (Source: U.S. Postal Service) Year, Rate, a. Create a scatter plot of the data. Find a linear model for the data, and add its graph to our scatter plot. According to this model, when will the rate for sending an epress mail package reach $9.00? b. Create a new table showing values for ln and ln and create a scatter plot of these transformed data. Use the method illustrated in Eample 7 in Section. to find a model for the transformed data, and add its graph to our scatter plot. According to this model, when will the rate for sending an epress mail package reach $9.00? c. Solve the model in part (b) for,and add its graph to our scatter plot in part (a). Which model better fits the original data? Which model will better predict future rates? Eplain.

37 Section. Eponential and Logarithmic Equations 5. Eercises VOCABULARY CHECK: Fill in the blanks.. To an equation in means to find all values of for which the equation is true.. To solve eponential and logarithmic equations, ou can use the following One-to-One and Inverse Properties. (a) a a if and onl if. (b) log a log a if and onl if. (c) a log a (d) log a a. An solution does not satisf the original equation. In Eercises 8, determine whether each -value is a solution (or an approimate solution) of the equation.. 7. (a) 5 (a) (b) (b). e 75 (a) (b) (c) e 5 ln 5.9. e 5 (a) (b) (c) 5 ln ln 5 ln log (a) (b) (c).. log 0 (a) (b) (c) ln 5.8 (a) (b) (c) ln 5.8 e ln.8 (a) (b) (c) e ln.8 In Eercises 9 0, solve for ln ln 0. ln ln e. e 7. ln 8. ln 7 9. log 0. log 5 In Eercises, approimate the point of intersection of the graphs of f and g. Then solve the equation f g algebraicall to verif our approimation.. f. f 7 g 8 8. f log. f ln g g f f g 8 8 g 9 8 g 0 8 g 8 f f g 8 8

38 5 Chapter Eponential and Logarithmic Functions In Eercises 5, solve the eponential equation algebraicall. Approimate the result to three decimal places. 5. e e. e e 8 7. e e 8. e e e 0. e 9. e t t e 8. e e e e 5 5. e e e e 5e e e e 9e e 0 e e e t t t t 0 In Eercises 7 7, use a graphing utilit to graph and solve the equation. Approimate the result to the nearest thousandth. Verif our result algebraicall. 7. e 5 8. e e e 7. e 0.09t 7. e e 0.5t e.7 9 In Eercises 75 0, solve the logarithmic equation algebraicall. Approimate the result to three decimal places. 75. ln 7. ln 77. ln. 78. ln 79. log 80. log z 8. ln ln 7 8. ln 8. ln ln 5 8. ln log log ln ln 90. ln ln 9. ln ln 9. ln ln 9. ln 5 ln ln 9. ln ln ln 95. log log 9. log log 97. log log log 98. log log log 99. log log 00. log log 8 0. log 8 log 0. log log In Eercises 0 0, use a graphing utilit to graph and solve the equation. Approimate the result to three decimal places. Verif our result algebraicall e 05. ln ln 0 Compound Interest In Eercises 07 and 08, $500 is invested in an account at interest rate r, compounded continuousl. Find the time required for the amount to (a) double and (b) triple. 07. r r Demand The demand equation for a microwave oven is given b p e Find the demand for a price of (a) p $50 and (b) p $ Demand The demand equation for a hand-held electronic organizer is p 5000 Find the demand for a price of (a) p $00 and (b) p $00.. Forest Yield The ield V (in millions of cubic feet per acre) for a forest at age t ears is given b V.7e 8. t. e (a) Use a graphing utilit to graph the function. (b) Determine the horizontal asmptote of the function. Interpret its meaning in the contet of the problem. (c) Find the time necessar to obtain a ield of. million cubic feet.

39 Section. Eponential and Logarithmic Equations 55. Trees per Acre The number N of trees of a given species per acre is approimated b the model N , 5 0 where is the average diameter of the trees (in inches) feet above the ground. Use the model to approimate the average diameter of the trees in a test plot when N.. Medicine The number of hospitals in the United States from 995 to 00 can be modeled b ln t, 5 t where t represents the ear, with t 5 corresponding to 995. During which ear did the number of hospitals reach 5800? (Source: Health Forum). Sports The number of dail fee golf facilities in the United States from 995 to 00 can be modeled b ln t, 5 t where t represents the ear, with t 5 corresponding to 995. During which ear did the number of dail fee golf facilities reach 9000? (Source: National Golf Foundation) 5. Average Heights The percent m of American males between the ages of 8 and who are no more than inches tall is modeled b m and the percent f of American females between the ages of 8 and who are no more than inches tall is modeled b f (Source: U.S. National Center for Health Statistics) (a) Use the graph to determine an horizontal asmptotes of the graphs of the functions. Interpret the meaning in the contet of the problem. Percent of population e e (b) What is the average height of each se?. Learning Curve In a group project in learning theor, a mathematical model for the proportion P of correct responses after n trials was found to be P 0.8 e 0.n. f() m() Height (in inches) (a) Use a graphing utilit to graph the function. (b) Use the graph to determine an horizontal asmptotes of the graph of the function. Interpret the meaning of the upper asmptote in the contet of this problem. (c) After how man trials will 0% of the responses be correct? 7. Automobiles Automobiles are designed with crumple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer g s the crash victims eperience. (One g is equal to the acceleration due to gravit. For ver short periods of time, humans have withstood as much as 0 g s.) In crash tests with vehicles moving at 90 kilometers per hour, analsts measured the numbers of g s eperienced during deceleration b crash dummies that were permitted to move meters during impact. The data are shown in the table. A model for the data is given b ln.9 where is the number of g s. (a) Complete the table using the model Model It g s (b) Use a graphing utilit to graph the data points and the model in the same viewing window. How do the compare? (c) Use the model to estimate the distance traveled during impact if the passenger deceleration must not eceed 0 g s. (d) Do ou think it is practical to lower the number of g s eperienced during impact to fewer than? Eplain our reasoning.

40 5 Chapter Eponential and Logarithmic Functions 8. 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 the data is given b T 0 7 h. The graph of this model is shown in the figure. (a) Use the graph to identif the horizontal asmptote of the model and interpret the asmptote in the contet of the problem. (b) Use the model to approimate the time when the temperature of the object was 00 C. Temperature (in degrees Celsius) Snthesis T Hour, h Temperature, T Hour True or False? In Eercises 9, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justif our answer. 9. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. 0. The logarithm of the sum of two numbers is equal to the product of the logarithms of the numbers.. The logarithm of the difference of two numbers is equal to the difference of the logarithms of the numbers. h. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers.. Think About It Is it possible for a logarithmic equation to have more than one etraneous solution? Eplain.. Finance You are investing P dollars at an annual interest rate of r, compounded continuousl, for t ears. Which of the following would result in the highest value of the investment? Eplain our reasoning. (a) Double the amount ou invest. (b) Double our interest rate. (c) Double the number of ears. 5. Think About It Are the times required for the investments in Eercises 07 and 08 to quadruple twice as long as the times for them to double? Give a reason for our answer and verif our answer algebraicall.. Writing Write two or three sentences stating the general guidelines that ou follow when solving (a) eponential equations and (b) logarithmic equations. Skills Review In Eercises 7 0, simplif the epression In Eercises, sketch a graph of the function..... f 9 f 8 g,, g,, In Eercises 5 8, evaluate the logarithm using the change-of-base formula. Approimate our result to three decimal places. 5. log 9. log 7. log 5 8. log 8 < 0 0 >

41 Section.5 Eponential and Logarithmic Models 57.5 Eponential and Logarithmic Models What ou should learn 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. Wh ou should learn it Eponential growth and deca models are often used to model the population of a countr. For instance, in Eercise on page 5, ou will use eponential growth and deca models to compare the populations of several countries. Introduction The five most common tpes of mathematical models involving eponential functions and logarithmic functions are as follows.. Eponential growth model:. Eponential deca model: ae b, ae b, b > 0 b > 0. Gaussian model:. Logistic growth model: 5. Logarithmic models: ae ( b) c a be r a b ln, a b log The basic shapes of the graphs of these functions are shown in Figure.9. = e = e = e EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL GAUSSIAN MODEL = + ln = + log = + e 5 LOGISTIC GROWTH MODEL NATURAL LOGARITHMIC MODEL COMMON LOGARITHMIC MODEL FIGURE.9 Alan Becker/Gett Images You can often gain quite a bit of insight into a situation modeled b an eponential or logarithmic function b identifing and interpreting the function s asmptotes. Use the graphs in Figure.9 to identif the asmptotes of the graph of each function.

42 58 Chapter Eponential and Logarithmic Functions Eponential Growth and Deca Eample Digital Television Households (in millions) D FIGURE.0 D Digital Television 5 7 Year ( 00) Digital Television t Estimates of the numbers (in millions) of U.S. households with digital television from 00 through 007 are shown in the table. The scatter plot of the data is shown in Figure.0. (Source: emarketer) Year An eponential growth model that approimates these data is given b D 0.9e 0.7t, t 7 Households Households (in millions) FIGURE. 5 7 Year ( 00) t where D is the number of households (in millions) and t represents 00. Compare the values given b the model with the estimates shown in the table. According to this model, when will the number of U.S. households with digital television reach 00 million? The following table compares the two sets of figures. The graph of the model and the original data are shown in Figure.. Year Households Model Technolog Some graphing utilities have an eponential regression feature that can be used to find eponential models that represent data. If ou have such a graphing utilit, tr using it to find an eponential model for the data given in Eample. How does our model compare with the model given in Eample? To find when the number of U.S. households with digital television will reach 00 million, let D 00 in the model and solve for t. 0.9e 0.7t D Write original model. 0.9e 0.7t 00 Let D 00. e 0.7t. Divide each side b 0.9. ln e 0.7t ln. Take natural log of each side. 0.7t.78 Inverse Propert t 0.0 Divide each side b 0.7. According to the model, the number of U.S. households with digital television will reach 00 million in 00. Now tr Eercise 5.

43 Section.5 Eponential and Logarithmic Models 59 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. Eample Modeling Population Growth Population Fruit Flies 00 (5, 50) =.e 0.59t 00 (, 00) (, 00) 5 Time (in das) FIGURE. 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 ae bt produces 00 ae b and 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 00 a e 00 e ln 00.. Substitute ln for b. Simplif. Inverse Propert Simplif. So, with a. and b ln 0.59, the eponential growth model is ln.e 0.59t 00 ae b. a 00 e b as shown in Figure.. This implies that, after 5 das, the population will be.e flies. Now tr Eercise 7. e b

44 0 Chapter Eponential and Logarithmic Functions Ratio 0 ( 0 ) 0 R t = 0 Carbon Dating R = 0 t = 5,700 5,000 5,000 Time (in ears) e t/8 t = 9,000 FIGURE. Eample t 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 Carbon dating model The graph of R is shown in Figure.. Note that R decreases as t increases. Carbon Dating Estimate the age of a newl discovered fossil in which the ratio of carbon to carbon is R 0. In the carbon dating model, substitute the given value of R following. 0 e t 8 R e t e t 8 0 Write original model. Let R 0. Multipl each side b 0. to obtain the ln e t 8 ln 0 Take natural log of each side. The carbon dating model in Eample assumed that the carbon to carbon ratio was one part in 0,000,000,000,000. Suppose an error in measurement occurred and the actual ratio was one part in 8,000,000,000,000. The fossil age corresponding to the actual ratio would then be approimatel 7,000 ears. Tr checking this result. t.0 8 t 8,9 Inverse Propert Multipl each side b 8. So, to the nearest thousand ears, the age of the fossil is about 9,000 ears. Now tr Eercise. The value of b in the eponential deca model ae bt determines the deca of radioactive isotopes. For instance, to find how much of an initial 0 grams of Ra isotope with a half-life of 599 ears is left after 500 ears, substitute this information into the model ae bt. 0 0e b 599 ln 599b Using the value of b found above and a 0, the amount left is 0e ln grams. b ln 599

45 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. The graph of a Gaussian model is called a bell-shaped curve. Tr graphing the normal distribution with a graphing utilit. Can ou see wh it is called a bell-shaped curve? For standard normal distributions, the model takes the form e. Section.5 Eponential and Logarithmic Models 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,. Eample SAT Scores In 00, the Scholastic Aptitude Test (SAT) math scores for college-bound seniors roughl followed the normal distribution given b 0.005e 58 5,99, where is the SAT score for mathematics. Sketch the graph of this function. From the graph, estimate the average SAT score. (Source: College Board) The graph of the function is shown in Figure.. On this bell-shaped curve, the maimum value of the curve represents the average score. From the graph, ou can estimate that the average mathematics score for college-bound seniors in 00 was 58. SAT Scores % of population Distribution = 58 FIGURE Score Now tr Eercise 7.

46 Chapter Eponential and Logarithmic Functions Logistic Growth Models Decreasing rate of growth Increasing rate of growth Some populations initiall have rapid growth, followed b a declining rate of growth, as indicated b the graph in Figure.5. One model for describing this tpe of growth pattern is the logistic curve given b the function a be r where is the population size and is the time. An eample is a bacteria culture that is initiall allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. A logistic growth curve is also called a sigmoidal curve. FIGURE.5 Eample 5 Spread of a Virus On a college campus of 5000 students, one student returns from vacation with a contagious and long-lasting flu virus. The spread of the virus is modeled b 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 b. Classes are canceled when the number infected is e 0.8t e 0.8t Flu Virus e 0.8t Students infected FIGURE. (0., 000) (5, 5) 8 0 Time (in das) t ln e 0.8t ln t ln t.5 ln t 0. So, after about 0 das, at least 0% of the students will be infected, and the college will cancel classes. The graph of the function is shown in Figure.. Now tr Eercise 9.

47 Logarithmic Models Section.5 Eponential and Logarithmic Models BAY ISMOYO/AFP/Gett Images On December, 00, an earthquake of magnitude 9.0 struck northern Sumatra and man other Asian countries. This earthquake caused a deadl tsunami and was the fourth largest earthquake in the world since 900. Eample Magnitudes of Earthquakes On the Richter scale, the magnitude R of an earthquake of intensit I is given b R log I I 0 where I 0 is the minimum intensit used for comparison. Find the intensities per unit of area for each earthquake. (Intensit is a measure of the wave energ of an earthquake.) a. Northern Sumatra in 00: R 9.0 b. Southeastern Alaska in 00: R.8 a. Because I 0 and R 9.0, ou have 9.0 log I Substitute for I 0 and 9.0 for R. t Year Population, P log I Eponentiate each side. I ,000,000. Inverse Propert b. For R.8, ou have I.8 log Substitute for I and.8 for R log I Eponentiate each side. I 0.8,0,000. Inverse Propert Note that an increase of. units on the Richter scale (from.8 to 9.0) represents an increase in intensit b a factor of,000,000,000,0, In other words, the intensit of the earthquake in Sumatra was about 58 times greater than that of the earthquake in Alaska. Now tr Eercise 5. W RITING ABOUT MATHEMATICS Comparing Population Models The populations P (in millions) of the United States for the census ears from 90 to 000 are shown in the table at the left. Least squares regression analsis gives the best quadratic model for these data as P.08t 9.07t 8.8, and the best eponential model for these data as P 8.77e 0.t. Which model better fits the data? Describe how ou reached our conclusion. (Source: U.S. Census Bureau)