3.4 EXPONENTIAL AND LOGARITHMIC EQUATIONS

Transcription

1 Chapter 3 Eponential and Logarithmic Functions 3. EXPONENTIAL AND LOGARITHMIC EQUATIONS What ou should learn Solve simple eponential and logarithmic equations. Solve more complicated eponential equations. Solve more complicated logarithmic equations. Use eponential and logarithmic equations to model and solve real-life problems. Wh ou should learn it Eponential and logarithmic equations are used to model and solve life science applications. For instance, in Eercise 13 on page 53, an eponential function is used to model the number of trees per acre given the average diameter of the trees. Introduction So far in this chapter, ou have studied the definitions, graphs, and properties of eponential and logarithmic functions. In this section, ou will stud procedures for solving equations involving these eponential and logarithmic functions. There are two basic strategies for solving eponential or logarithmic equations. The first is based on the One-to-One Properties and was used to solve simple eponential and logarithmic equations in Sections 3.1 and 3.. The second is based on the Inverse Properties. For a > 0 and a 1, the following properties are true for all and for which log a and log a are defined. One-to-One Properties a a if and onl if. log a log a if and onl if. Inverse Properties a log a log a a Eample 1 Solving Simple Equations James Marshall/Corbis Original Rewritten Equation Equation Propert a One-to-One b. ln ln 3 0 ln ln 3 3 One-to-One c. 3 3 One-to-One d. e 7 ln e ln 7 ln 7 Inverse e. ln 3 e ln e 3 e 3 Inverse f. log 1 10 log Inverse g. log 3 3 log Inverse Now tr Eercise 17. The strategies used in Eample 1 are summarized as follows. Strategies for Solving Eponential and Logarithmic Equations 1. 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. 3. Rewrite a logarithmic equation in eponential form and appl the Inverse Propert of eponential functions.

2 Section 3. Eponential and Logarithmic Equations 5 Solving Eponential Equations Eample Solving Eponential Equations Solve each equation and approimate the result to three decimal places, if necessar. a. e e 3 b. 3 Another wa to solve Eample (b) is b taking the natural log of each side and then appling the Power Propert, as follows. 3 1 ln ln 1 ln ln 1 ln 1 ln As ou can see, ou obtain the same result as in Eample (b). a. e e One-to-One Propert Write in general form. Factor. Set 1st factor equal to 0. Set nd factor equal to 0. The solutions are 1and. Check these in the original equation. b. 3 1 log log 1 Divide each side b 3. Take log (base ) of each side. Change-of-base formula The solution is log Check this in the original equation. Now tr Eercise 9. In Eample (b), the eact solution is log 1 and the approimate solution is 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 log 1 ln 1 ln Solving an Eponential Equation Solve e 5 60 and approimate the result to three decimal places. Remember that the natural logarithmic function has a base of e. e 5 60 e 55 ln e ln 55 ln Subtract 5 from each side. Take natural log of each side. The solution is ln Check this in the original equation. Now tr Eercise 55.

3 6 Chapter 3 Eponential and Logarithmic Functions Eample Solving an Eponential Equation Solve 3 t 5 11 and approimate the result to three decimal places. 3 t t t 5 15 Add to each side. Divide each side b. Remember that to evaluate a logarithm such as log 3 7.5, ou need to use the change-of-base formula. log ln 7.5 ln log 3 3 t 5 log 3 15 t 5 log 3 15 t 5 log t 5 1 log t 3.17 Take log (base 3) of each side. Add 5 to each side. Divide each side b. Use a calculator. The solution is t 5 1 log Check this in the original equation. Now tr Eercise 57. When an equation involves two or more eponential epressions, ou can still use a procedure similar to that demonstrated in Eamples, 3, and. However, the algebra is a bit more complicated. Eample 5 Solving an Eponential Equation of Quadratic Tpe Solve e 3e 0. Algebraic e 3e 0 e 3e 0 e e 1 0 e 0 ln e Write in quadratic form. Factor. Set 1st factor equal to 0. Set nd factor equal to 0. The solutions are ln and 0. Check these in the original equation. Now tr Eercise 59. Graphical Use a graphing utilit to graph e 3e. 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 3.8, ou can see that the zeros occur at 0 and at So, the solutions are 0 and = e 3e FIGURE 3.8

4 Section 3. Eponential and Logarithmic Equations 7 Solving Logarithmic Equations To solve a logarithmic equation, ou can write it in eponential form. ln 3 Logarithmic form e ln e 3 Eponentiate each side. e 3 Eponential form This procedure is called eponentiating each side of an equation. Eample 6 Solving Logarithmic Equations WARNING / CAUTION Remember to check our solutions in the original equation when solving equations to verif that the answer is correct and to make sure that the answer lies in the domain of the original equation. a. ln e ln e e Original equation Eponentiate each side. b. log log Original equation One-to-One Propert Add and 1 to each side. Divide each side b. c. log log 6 5 log 6 Original equation log log Now tr Eercise 83. Quotient Propert of Logarithms One-to-One Propert Cross multipl. Isolate. Divide each side b 7. Eample 7 Solving a Logarithmic Equation Solve 5 ln and approimate the result to three decimal places. Algebraic 5 ln ln 1 ln 1 e ln e 1 e Now tr Eercise 93. Subtract 5 from each side. Divide each side b. Eponentiate each side. Use a calculator. Graphical Use a graphing utilit to graph 1 5 ln and in the same viewing window. Use the intersect feature or the zoom and trace features to approimate the intersection point, as shown in Figure 3.9. So, the solution is = 1 = 5 + ln FIGURE 3.9

5 8 Chapter 3 Eponential and Logarithmic Functions Eample 8 Solving a Logarithmic Equation Solve log 5 3. log 5 3 log log Divide each side b. Eponentiate each side (base 5). Divide each side b 3. 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 3. Check this in the original equation. Now tr Eercise 97. 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 1. Algebraic log 5 log 1 log log Product Propert of Logarithms Eponentiate each side (base 10). Write in general form. Factor. 5 0 Set 1st 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 109. Graphical Use a graphing utilit to graph 1 log 5 log 1 and in the same viewing window. From the graph shown in Figure 3.30, 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 = log 5 + log( 1) FIGURE 3.30 = 9 In Eample 9, the domain of log 5 is > 0 and the domain of log 1 is > 1, so the domain of the original equation is > 1. Because the domain is all real numbers greater than 1, the solution is etraneous. The graph in Figure 3.30 verifies this conclusion.

6 C WASHINGTON SER IES 1993 C 31 Section 3. Eponential and Logarithmic Equations 9 Applications Eample 10 Doubling an Investment You have deposited $500 in an account that pas 6.75% interest, compounded continuousl. How long will it take our mone to double? Using the formula for continuous compounding, ou can find that the balance in the account is A Pe rt A 500e t. To find the time required for the balance to double, let A 0 and solve the resulting equation for t. 500e t 0 e t ln e t ln t ln t ln t 10.7 Let A 0. Divide each side b 500. Take natural log of each side. Divide each side b Use a calculator. The balance in the account will double after approimatel 10.7 ears. This result is demonstrated graphicall in Figure A Doubling an Investment Account balance (in dollars) FIGURE 3.31 THE UNITED STATES OF AMERICA (0, 500) Now tr Eercise 117. A WASHINGTON, D.C (10.7, 0) A = 500e t 6 8 Time (in ears) 10 t In Eample 10, an approimate answer of 10.7 ears is given. Within the contet of the problem, the eact solution, ln ears, does not make sense as an answer.

7 50 Chapter 3 Eponential and Logarithmic Functions Sales (in billions) FIGURE 3.3 Retail Sales of e-commerce Companies Year (1 00) t Eample 11 Retail Sales The retail sales (in billions) of e-commerce companies in the United States from 00 through 007 can be modeled b ln t, 1 t 17 where t represents the ear, with t 1 corresponding to 00 (see Figure 3.3). During which ear did the sales reach $108 billion? (Source: U.S. Census Bureau) ln t ln t ln t 657 Substitute 108 for. Add 59 to each side. ln t 657 Divide each side b e ln t e Eponentiate each side. t e t 16 Use a calculator. The solution is t 16. Because t 1 represents 00, it follows that the sales reached $108 billion in 006. Now tr Eercise 133. CLASSROOM DISCUSSION Analzing Relationships Numericall Use a calculator to fill in the table row-brow. Discuss the resulting pattern. What can ou conclude? Find two equations that summarize the relationships ou discovered. e ln e ln e ln

8 Section 3. Eponential and Logarithmic Equations EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks. 1. 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 3. To solve eponential and logarithmic equations, ou can use the following strategies. (a) Rewrite the original equation in a form that allows the use of the Properties of eponential or logarithmic functions. (b) Rewrite an eponential equation in form and appl the of functions. (c) Rewrite a logarithmic equation in form and appl the of functions.. An solution does not satisf the original equation. SKILLS AND APPLICATIONS In Eercises 5 1, determine whether each -value is a solution (or an approimate solution) of the equation (a) 5 (a) 1 (b) (b) 7. 3e e 1 60 (a) e 5 (a) 1 ln 15 (b) ln 5 (b) (c) 1.19 (c) ln log log 3 10 (a) (a) 101 (b) (b) 17 (c) 6 3 (c) ln ln (a) 1 3 ln 5.8 (a) 1 e 3.8 (b) 1 3 e 5.8 (b) (c) (c) 1 ln f 6. g f log 3 8. g g 1 f f g f 7 g 9 8 f ln g g 8 f f g In Eercises 13, solve for ln ln ln ln e 0. e 1. ln 1. log 3. log 3. log 5 1 In Eercises 5 8, approimate the point of intersection of the graphs of f and g. Then solve the equation f g algebraicall to verif our approimation. In Eercises 9 70, solve the eponential equation algebraicall. Approimate the result to three decimal places. 9. e e 30. e e e 3 e 3. e e e e e t t

9 5 Chapter 3 Eponential and Logarithmic Functions e e e e e e e e e 5e e 3e 0 6. e 9e e 1 e e e t t t t 30 In Eercises 71 80, use a graphing utilit to graph and solve the equation. Approimate the result to three decimal places. Verif our result algebraicall e e e e e 0.09t e e 0.15t e.7 9 In Eercises 81 11, solve the logarithmic equation algebraicall. Approimate the result to three decimal places. 81. ln 3 8. ln ln ln ln ln log log 3z 89. 3ln ln ln 1 9. ln ln 5 6 ln 10 ln ln 1 6 log log 6 11 ln ln 1 ln ln 1 1 ln ln 1 ln ln ln 5 ln 1 ln ln 1 ln ln 105. log 3 log 106. log 3 log log log log 108. log log log log log log 3 log log 8 log log log 1 In Eercises , use a graphing utilit to graph and solve the equation. Approimate the result to three decimal places. Verif our result algebraicall ln ln ln ln 1 ln COMPOUND INTEREST In Eercises , $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 r r r r In Eercises 11 18, solve the equation algebraicall. Round the result to three decimal places. Verif our answer using a graphing utilit. 11. e e 0 1. e e e e 0 1. e e ln ln DEMAND The demand equation for a limited edition coin set is p 0 1 Find the demand for a price of (a) p $ and (b) p $ DEMAND The demand equation for a hand-held electronic organizer is p e e ln 0 ln 1 0 Find the demand for a price of (a) p $600 and (b) p $00.

10 Section 3. Eponential and Logarithmic Equations FOREST YIELD The ield V (in millions of cubic feet per acre) for a forest at age t ears is given b V 6.7e 8.1 t. (a) Use a graphing utilit to graph the function. (b) Determine the horizontal asmptote of the function. Interpret its meaning in the contet of the problem. (c) Find the time necessar to obtain a ield of 1.3 million cubic feet. 13. 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) 3 feet above the ground. Use the model to approimate the average diameter of the trees in a test plot when N U.S. CURRENCY The values (in billions of dollars) of U.S. currenc in circulation in the ears 000 through 007 can be modeled b 51 ln t, 10 t 17, where t represents the ear, with t 10 corresponding to 000. During which ear did the value of U.S. currenc in circulation eceed $690 billion? (Source: Board of Governors of the Federal Reserve Sstem) 13. MEDICINE The numbers of freestanding ambulator care surger centers in the United States from 000 through 007 can be modeled b 875 where t represents the ear, with t 0 corresponding to 000. (Source: Verispan) (a) Use a graphing utilit to graph the model. (b) Use the trace feature of the graphing utilit to estimate the ear in which the number of surger centers eceeded AVERAGE HEIGHTS The percent m of American males between the ages of 18 and who are no more than inches tall is modeled b m and the percent f of American females between the ages of 18 and who are no more than inches tall is modeled b f e t, 0 t 7 1 e e (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 f() Height (in inches) (b) What is the average height of each se? 136. 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 e 0.n. (a) Use a graphing utilit to graph the function. (b) Use the graph to determine an horizontal asmptotes of the graph of the function. Interpret the meaning of the upper asmptote in the contet of this problem. (c) After how man trials will 60% of the responses be correct? 137. 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 36.9, where is the number of g s. (a) Complete the table using the model g s m()

11 5 Chapter 3 Eponential and Logarithmic Functions (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 30 g s. (d) Do ou think it is practical to lower the number of g s eperienced during impact to fewer than 3? Eplain our reasoning DATA ANALYSIS An object at a temperature of 160 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 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 C. Temperature (in degrees Celsius) EXPLORATION Hour, h T Temperature, T Hour TRUE OR FALSE? In Eercises 139 1, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justif our answer The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. h 10. The logarithm of the sum of two numbers is equal to the product of the logarithms of the numbers. 11. The logarithm of the difference of two numbers is equal to the difference of the logarithms of the numbers. 1. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers. 13. THINK ABOUT IT Is it possible for a logarithmic equation to have more than one etraneous solution? Eplain. 1. 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. 15. THINK ABOUT IT Are the times required for the investments in Eercises to quadruple twice as long as the times for them to double? Give a reason for our answer and verif our answer algebraicall. 16. The effective ield of a savings plan is the percent increase in the balance after 1 ear. Find the effective ield for each savings plan when $0 is deposited in a savings account. Which savings plan has the greatest effective ield? Which savings plan will have the highest balance after 5 ears? (a) 7% annual interest rate, compounded annuall (b) 7% annual interest rate, compounded continuousl (c) 7% annual interest rate, compounded quarterl (d) 7.5% annual interest rate, compounded quarterl 17. GRAPHICAL ANALYSIS Let f log a and g a, where a > 1. (a) Let a 1. and use a graphing utilit to graph the two functions in the same viewing window. What do ou observe? Approimate an points of intersection of the two graphs. (b) Determine the value(s) of a for which the two graphs have one point of intersection. (c) Determine the value(s) of a for which the two graphs have two points of intersection. 18. CAPSTONE Write two or three sentences stating the general guidelines that ou follow when solving (a) eponential equations and (b) logarithmic equations.