Chapter 3. Exponential and Logarithmic Functions. Selected Applications
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1 Chapter Eponential and Logarithmic Functions. Eponential Functions and Their Graphs. Logarithmic Functions and Their Graphs. Properties of Logarithms. Solving Eponential and Logarithmic Equations.5 Eponential and Logarithmic Models.6 Nonlinear Models 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. Radioactive Deca, Eercises 6 and 68, page 9 Sound Intensit, Eercise 95, page 5 Home Mortgage, Eercise 96, page 5 Comparing Models, Eercise 9, page Forestr, Eercise 8, page IQ Scores, Eercise, page Newton s Law of Cooling, Eercises 5 and 5, page 6 Elections, Eercise, page f() = e f() = e f() = e f () = ln Eponential and logarithmic functions are called transcendental functions because these functions are not algebraic. In Chapter, ou will learn about the inverse relationship between eponential and logarithmic functions, how to graph these functions, how to solve eponential and logarithmic equations, and how to use these functions in real-life applications. Denis O Regan/Corbis The relationship between the number of decibels and the intensit of a sound can be modeled b a logarithmic function. A rock concert at a stadium has a decibel rating of decibels. Sounds at this level can cause gradual hearing loss. 8
2 8 Chapter Eponential and Logarithmic Functions. Eponential Functions and Their Graphs 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 >, a, and is an real number. Note that in the definition of an eponential function, the base a is ecluded because it ields f. This is a constant function, not an eponential function. You have alread evaluated a for integer and rational values of. For eample, ou know that 6 and. However, to evaluate for an real number, ou need to interpret forms with irrational eponents. For the purposes of this tet, it is sufficient to think of a where.56 as the number that has the successivel closer approimations a., a., a., a., a.,.... Eample shows how to use a calculator to evaluate eponential functions. Eample Evaluating Eponential Functions Use a calculator to evaluate each function at the indicated value of. Function Value a. b. c. f f f.6. Function Value Graphing Calculator Kestrokes Displa a. f... ENTER.669 b. f ENTER. c. f.6.6 ENTER.658 Now tr Eercise. > > > What ou should learn Recognize and evaluate eponential functions with base a. Graph eponential functions with base a. 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 are useful in modeling data that represents quantities that increase or decrease quickl. For instance, Eercise on page 95 shows how an eponential function is used to model the depreciation of a new vehicle. Sergio Piumatti TECHNOLOGY TIP 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 Graphs of Eponential Functions The graphs of all eponential functions have similar characteristics, as shown in Eamples,, and. Eample Graphs of a In the same coordinate plane, sketch the graph of each function b hand. a. f b. g The table below lists some values for each function. B plotting these points and connecting them with smooth curves, ou obtain the graphs shown in Figure.. Note that both graphs are increasing. Moreover, the graph of g is increasing more rapidl than the graph of f. Section. Eponential Functions and Their Graphs Figure. Now tr Eercise 5. Eample Graphs of a In the same coordinate plane, sketch the graph of each function b hand. a. F b. G The table below lists some values for each function. B plotting these points and connecting them with smooth curves, ou obtain the graphs shown in Figure.. Note that both graphs are decreasing. Moreover, the graph of G is decreasing more rapidl than the graph of F. Now tr Eercise. The properties of eponents can also be applied to real-number eponents. For review, these properties are listed below... a a a.. a a a ab a b a a a a a b a a a a a a b 6 Figure. STUDY TIP In Eample, note that the functions F and G can be rewritten with positive eponents. F G and
4 86 Chapter Eponential and Logarithmic Functions 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, as shown in Figure.. The graphs of G and g have the same relationship, as shown in Figure.. F() = f() = G() = g() = STUDY TIP Notice that the range of the eponential functions in Eamples and is,, which means that a > and a > for all values of. Figure. Figure. The graphs in Figures. and. are tpical of the graphs of the eponential functions f a and f a. The have one -intercept and one horizontal asmptote (the -ais), and the are continuous. Librar of Parent Functions: Eponential Function The eponential function f a, a >, is different from all the functions ou have studied so far because the variable is an eponent. A distinguishing characteristic of an eponential function is its rapid increase as increases for a >. Man real-life phenomena with patterns of rapid growth (or decline) can be modeled b eponential functions. The basic characteristics of the eponential function are summarized below. A review of eponential functions can be found in the Stud Capsules. Graph of f a, a > Domain:, Range:, Intercept:, Increasing on, -ais is a horizontal asmptote a as Continuous a Graph of f a, a > Domain:, Range:, Intercept:, Decreasing on, -ais is a horizontal asmptote a as Continuous Eploration Use a graphing utilit to graph a for a, 5, and in the same viewing window. (Use a viewing window in which and.) How do the graphs compare with each other? Which graph is on the top in the interval,? Which is on the bottom? Which graph is on the top in the interval,? Which is on the bottom? Repeat this eperiment with the graphs of b for b, 5, and. (Use a viewing window in which and.) What can ou conclude about the shape of the graph of b and the value of b? f() = a (, ) (, ) f() = a
5 Section. Eponential Functions and Their Graphs 8 In the following eample, the graph of a is used to graph functions of the form f b ± a c, where b and c are an real numbers. Eample 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.6. c. Because k f, the graph of k can be obtained b reflecting the graph of f in the -ais, as shown in Figure.. 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. Prerequisite Skills If ou have difficult with this eample, review shifting and reflecting of graphs in Section.. g() = + f() = f() = h() = Figure.5 Figure.6 f() = 5 j() = = f() = Eploration The following table shows some points on the graphs in Figure.5. The functions f and g are represented b Y and Y, respectivel. Eplain how ou can use the table to describe the transformation. k() = Figure. Figure.8 Now tr Eercise. Notice that the transformations in Figures.5,., and.8 keep the -ais as a horizontal asmptote, but the transformation in Figure.6 ields a new horizontal asmptote of. Also, be sure to note how the -intercept is affected b each transformation. The Natural Base e For man applications, the convenient choice for a base is the irrational number e
6 88 Chapter Eponential and Logarithmic Functions This number is called the natural base. The function f e is called the natural eponential function and its graph is shown in Figure.9. The graph of the eponential function has the same basic characteristics as the graph of the function f a (see page 86). Be sure ou see that for the eponential function f e, e is the constant , whereas is the variable. Eample 5 Figure.9 The Natural Eponential Function In Eample 5, ou will see that the number e can be approimated b the epression ( for large values of. (,, e e Approimation of the Number e ( ( 5 (, e) f() = e (, ) Evaluate the epression for several large values of to see that the values approach e.8888 as increases without bound. Eploration Use our graphing utilit to graph the functions e in the same viewing window. From the relative positions of these graphs, make a guess as to the value of the real number e. Then tr to find a number a such that the graphs of e and a are as close as possible. TECHNOLOGY SUPPORT For instructions on how to use the trace feature and the table feature, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. Graphical Use a graphing utilit to graph and e in the same viewing window, as shown in Figure.. Use the trace feature of the graphing utilit to verif that as increases, the graph of gets closer and closer to the line e. Numerical Use the table feature (in ask mode) of a graphing utilit to create a table of values for the function, beginning at and increasing the -values as shown in Figure.. = ( + ( = e Figure. Figure. Now tr Eercise. From the table, it seems reasonable to conclude that e as.
7 Section. Eponential Functions and Their Graphs 89 Eample 6 Evaluating the Natural Eponential Function Use a calculator to evaluate the function f e at each indicated value of. a. b..5 c.. Function Value Graphing Calculator Kestrokes Displa a. f e e ENTER.55 b. f.5 e.5 e.5 ENTER.85 c. f. e. e. ENTER.6 Now tr Eercise. Eample Graphing Natural Eponential Functions Sketch the graph of each natural eponential function. a. f e. b. g e.58 Eploration Use a graphing utilit to graph. Describe the behavior of the graph near. Is there a -intercept? How does the behavior of the graph near relate to the result of Eample 5? Use the table feature of a graphing utilit to create a table that shows values of for values of near, to help ou describe the behavior of the graph near this point. To sketch these two graphs, ou can use a calculator to construct a table of values, as shown below. f g After constructing the table, plot the points and connect them with smooth curves. Note that the graph in Figure. is increasing, whereas the graph in Figure. is decreasing. Use a graphing calculator to verif these graphs. 6 f() = e g() = e.58 Figure. Figure. Now tr Eercise.
8 9 Chapter Eponential and Logarithmic Functions Applications One of the most familiar eamples of eponential growth is that of an investment earning continuousl compounded interest. 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 P is P P Pr P r. This pattern of multipling the previous principal b r is then repeated each successive ear, as shown in the table. Time in ears Balance after each compounding P P P P r P P r P r r P r t P t P r t Eploration Use the formula A P r n nt to calculate the amount in an account when P $, r 6%, t ears, and the interest is compounded (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. 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. (The product nt represents the total number of times the interest will be compounded.) Then the interest rate per compounding period is r n, and the account balance after t ears is A P r n nt. Amount (balance) with n compoundings per ear If ou let the number of compoundings n increase without bound, the process approaches what is called continuous compounding. In the formula for n compoundings per ear, let m n r. This produces P m m rt P. m mrt A P n r nt STUDY TIP The interest rate r in the formula for compound interest should be written as a decimal. For eample, an interest rate of % would be written as r.. As m increases without bound, ou know from Eample 5 that approaches e. So, for continuous compounding, it follows that P m m rt P e rt m m and ou can write A Pe rt. This result is part of the reason that e is the natural choice for a base of an eponential function. 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
9 Eample 8 Finding the Balance for Compound Interest A total of $9 is invested at an annual interest rate of.5%, compounded annuall. Find the balance in the account after 5 ears. Section. Eponential Functions and Their Graphs 9 Algebraic In this case, P 9, r.5%.5, n, t 5. Using the formula for compound interest with n compoundings per ear, ou have A P r n nt $,8.6. Formula for compound interest Substitute for P, r, n, and t. Simplif. Use a calculator. So, the balance in the account after 5 ears will be about $,8.6. Graphical Substitute the values for P, r, and n into the formula for compound interest with n compoundings per ear as follows. A P r n nt 9.5 t 9.5 t Formula for compound interest Substitute for P, r, and n. Simplif. Use a graphing utilit to graph 9.5. Using the value feature or the zoom and trace features, ou can approimate the value of when 5 to be about,8.6, as shown in Figure.. So, the balance in the account after 5 ears will be about $,8.6., Now tr Eercise 5. Figure. Eample 9 Finding Compound Interest A total of $, is invested at an annual interest rate of %. Find the balance after ears if the interest is compounded (a) quarterl and (b) continuousl. a. For quarterl compoundings, n. So, after ears at %, the balance is A P r n nt,. () b. For continuous compounding, the balance is A Pe rt,e.() $, $,5.9. Note that a continuous-compounding account ields more than a quarterlcompounding account. Now tr Eercise 55.
10 9 Chapter Eponential and Logarithmic Functions Eample Radioactive Deca Let represent a mass, in grams, of radioactive strontium 9 Sr, whose half-life is 9 ears. The quantit of strontium present after t ears is a. What is the initial mass (when t )? b. How much of the initial mass is present after 8 ears? t 9. Algebraic a. Write original equation. t 9 9 Substitute for t. Simplif. So, the initial mass is grams. b. Write original equation. t Substitute 8 for t. Simplif. Use a calculator. So, about.8 grams is present after 8 ears. Now tr Eercise 6. Graphical Use a graphing utilit to graph a. Use the value feature or the zoom and trace features of the graphing utilit to determine that the value of when is, as shown in Figure.5. So, the initial mass is grams. b. Use the value feature or the zoom and trace features of the graphing utilit to determine that the value of when 8 is about.8, as shown in Figure.6. So, about.8 grams is present after 8 ears Figure.5 Figure.6 5 Eample Population Growth The approimate number of fruit flies in an eperimental population after t hours is given b Q t e.t, where t. a. Find the initial number of fruit flies in the population. b. How large is the population of fruit flies after hours? c. Graph Q. a. To find the initial population, evaluate Q t when t. Q(t) = e.t, t Q e.() e flies b. After hours, the population size is Q e. e.6 flies. c. The graph of Q is shown in Figure.. Now tr Eercise 69. Figure. 8
11 Section. Eponential Functions and Their Graphs 9. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. Polnomial and rational functions are eamples of functions.. Eponential and logarithmic functions are eamples of nonalgebraic functions, also called functions.. The eponential function 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 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 annual interest rate r compounded continuousl, ou can use the formula. In Eercises, use a calculator to evaluate the function at the indicated value of. Round our result to three decimal places. Function Value. f f.. g 5. h 8.6 In Eercises 5, graph the eponential function b hand. Identif an asmptotes and intercepts and determine whether the graph of the function is increasing or decreasing. 5. g 5 6. f. f 5 8. h 9. h 5. g. g 5. f Librar of Parent Functions In Eercises 6, use the graph of to match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) (b) (d) 5 5. f. f 5. f 6. f In Eercises, use the graph of f to describe the transformation that ields the graph of g f, g 5 f, g 5 f 5, g 5 f., g. 5.. f, g In Eercises 6, use a calculator to evaluate the function at the indicated value of. Round our result to the nearest thousandth f, g Function Value f e 9. f e g 5e. h 5.5e In Eercises, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. Identif an asmptotes of the graph. f 5 f f 6. f. f. f
12 9 Chapter Eponential and Logarithmic Functions. f e 8. s t e.t 9. f e. f e.5. f e 5. g e. s t e.t. g e In Eercises 5 8, use a graphing utilit to (a) graph the function and (b) find an asmptotes numericall b creating a table of values for the function f 6. g e.5 e.5 6. f 6 8. f e. e. In Eercises 9 and 5, use a graphing utilit to find the point(s) of intersection, if an, of the graphs of the functions. Round our result to three decimal places. 9. e.5 5. e. 5 In Eercises 5 and 5, (a) use a graphing utilit to graph the function, (b) use the graph to find the open intervals on which the function is increasing and decreasing, and (c) approimate an relative maimum or minimum values. 5. f e 5. f e Compound Interest In Eercises 5 56, complete the table to determine the balance A for P dollars invested at rate r for t ears and compounded n times per ear. n 65 Continuous A 5. P $5, r.5%, t ears 5. P $, r 6%, t ears 55. P $5, r %, t ears 56. P $, r %, t ears Compound Interest In Eercises 5 6, complete the table to determine the balance A for $, invested at a rate r for t ears, compounded continuousl. t 5 A,5 5. r % 58. r 6% 59. r.5% 6. r.5% Annuit In Eercises 6 6, find the total amount A of an annuit after n months using the annuit formula A P r/ n r/ where P is the amount deposited ever month earning r% interest, compounded monthl. 6. P $5, r %, n 8 months 6. P $, r 9%, n 6 months 6. P $, r 6%, n months 6. P $5, r %, n months 65. Demand The demand function for a product is given b p 5 where p is the price and is the number of units. (a) Use a graphing utilit to graph the demand function for > and p >. (b) Find the price p for a demand of 5 units. (c) Use the graph in part (a) to approimate the highest price that will still ield a demand of at least 6 units. Verif our answers to parts (b) and (c) numericall b creating a table of values for the function. 66. Compound Interest There are three options for investing $5. The first earns % compounded annuall, the second earns % compounded quarterl, and the third earns % compounded continuousl. (a) Find equations that model each investment growth and use a graphing utilit to graph each model in the same viewing window over a -ear period. (b) Use the graph from part (a) to determine which investment ields the highest return after ears. What is the difference in earnings between each investment? 6. Radioactive Deca Let Q represent a mass, in grams, of radioactive radium 6 Ra, whose half-life is 599 ears. The quantit of radium present after t ears is given b Q 5 t 599. e. (a) Determine the initial quantit (when t ). (b) Determine the quantit present after ears. (c) Use a graphing utilit to graph the function over the interval t to t 5. (d) When will the quantit of radium be grams? Eplain. 68. Radioactive Deca Let Q represent a mass, in grams, of carbon C, whose half-life is 55 ears. The quantit present after t ears is given b Q t 55. (a) Determine the initial quantit (when t ). (b) Determine the quantit present after ears. (c) Sketch the graph of the function over the interval t to t,.
13 Section. Eponential Functions and Their Graphs Bacteria Growth A certain tpe of bacteria increases according to the model P t e.9t, where t is the time in hours. (a) Use a graphing utilit to graph the model. (b) Use a graphing utilit to approimate P, P 5, and P. (c) Verif our answers in part (b) algebraicall.. Population Growth The projected populations of California for the ears 5 to can be modeled b where P is the population (in millions) and t is the time (in ears), with t 5 corresponding to 5. (Source: U.S. Census Bureau) (a) Use a graphing utilit to graph the function for the ears 5 through. (b) Use the table feature of a graphing utilit to create a table of values for the same time period as in part (a). (c) According to the model, when will the population of California eceed 5 million?. Inflation If the annual rate of inflation averages % over the net ears, the approimate cost C of goods or services during an ear in that decade will be modeled b C t P. 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. (a) Use a graphing utilit to graph the function. (b) Use the graph in part (a) to approimate the price of an oil change ears from now. (c) Verif our answer in part (b) algebraicall.. Depreciation In earl 6, a new Jeep Wrangler Sport Edition had a manufacturer s suggested retail price of $,9. After t ears the Jeep s value is given b (Source: DaimlerChrsler Corporation) (a) Use a graphing utilit to graph the function. (b) Use a graphing utilit to create a table of values that shows the value V for t to t ears. (c) According to the model, when will the Jeep have no value? Snthesis True or False? In Eercises and, determine whether the statement is true or false. Justif our answer.. f is not an eponential function.. P.6e.9t V t,9 t. e,8 99,99 5. Librar of Parent Functions Determine which equation(s) ma be represented b the graph shown. (There ma be more than one correct answer.) (a) e (b) e (c) e (d) e 6. Eploration Use a graphing utilit to graph e and each of the functions,,, and 5 in the same viewing window. (a) Which function increases at the fastest rate for large values of? (b) Use the result of part (a) to make a conjecture about the rates of growth of and n e, where n is a natural number and is large. (c) Use the results of parts (a) and (b) to describe what is implied when it is stated that a quantit is growing eponentiall.. Graphical Analsis Use a graphing utilit to graph f.5 and g e.5 in the same viewing window. What is the relationship between f and g as increases without bound? 8. Think About It Which functions are eponential? Eplain. (a) (b) (c) (d) Think About It In Eercises 9 8, place the correct smbol < or > between the pair of numbers e e 5 5 Skills Review In Eercises 8 86, determine whether the function has an inverse function. If it does, find f. 8. f 5 8. f f f 6 In Eercises 8 and 88, sketch the graph of the rational function. 8. f 88. f 89. Make a Decision To work an etended application analzing the population per square mile in the United States, visit this tetbook s Online Stud Center. (Data Source: U.S. Census Bureau)
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