Eponential and Logarithmic Functions 3 3. Eponential Functions and Their Graphs 3. Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3. Eponential and Logarithmic Equations 3.5 Eponential and Logarithmic Models In Mathematics Eponential functions involve a constant base and a variable eponent. The inverse of an eponential function is a logarithmic function. In Real Life Eponential and logarithmic functions are widel used in describing economic and phsical phenomena such as compound interest, population growth, memor retention, and deca of radioactive material. For instance, a logarithmic function can be used to relate an animal s weight and its lowest galloping speed. (See Eercise 95, page.) Juniors Bildarchiv / Alam IN CAREERS There are man careers that use eponential and logarithmic functions. Several are listed below. Astronomer Eample 7, page 0 Pschologist Eercise 3, page 53 Archeologist Eample 3, page 58 Forensic Scientist Eercise 75, page 5
Chapter 3 Eponential and Logarithmic Functions 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Monke Business Images Ltd/Stockbroker/PhotoLibrar 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 7 on page, an eponential function is used to model the concentration of a drug in the bloodstream. 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 3 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.35) as the number that has the successivel closer approimations a., a., a., a., a.,.... Eample and is an real number. Evaluating Eponential Functions Use a calculator to evaluate each function at the indicated value of. Function Value a. b. c. f f f 0. 3. 3 Solution Function Value Graphing Calculator Kestrokes Displa a. f 3. 3. 3. ENTER 0.9 b. f ENTER 0.337 c. f 3 0. 3. 3 ENTER 0.7580 Now tr Eercise 7. > > > 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.
Section 3. Eponential Functions and Their Graphs 7 Graphs of Eponential Functions The graphs of all eponential functions have similar characteristics, as shown in Eamples, 3, and 5. Eample Graphs of a You can review the techniques for sketching the graph of an equation in Section.. g() = f() = 3 3 FIGURE 3. G() = F() = 0 8 3 3 FIGURE 3. 0 8 In the same coordinate plane, sketch the graph of each function. a. f b. g Solution The table below lists some values for each function, and Figure 3. 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. Now tr Eercise 7. The table in Eample was evaluated b hand. You could, of course, use a graphing utilit to construct tables with even more values. Eample 3 Graphs of a In the same coordinate plane, sketch the graph of each function. a. F b. G Solution The table below lists some values for each function, and Figure 3. 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 9. In Eample 3, note that b using one of the properties of eponents, the functions F and G can be rewritten with positive eponents. F 3 0 8 and 0 3 8 G
8 Chapter 3 Eponential and Logarithmic Functions Comparing the functions in Eamples and 3, 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 3. and 3. 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 3.3 and 3.. Notice that the range of an eponential function is 0,, which means that a > 0 for all values of. = a (0, ) Graph of a, a > Domain:, Range: 0, -intercept: 0, Increasing -ais is a horizontal asmptote a 0 as. Continuous FIGURE 3.3 (0, ) = a Graph of a, a > Domain:, Range: 0, -intercept: 0, Decreasing -ais is a horizontal asmptote a 0 as. Continuous FIGURE 3. From Figures 3.3 and 3., 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 3 3 3 Original equation 9 3 One-to-One Propert Solve for. b. 8 3 3 Now tr Eercise 5.
Section 3. Eponential Functions and Their Graphs 9 In the following eample, notice how the graph of a the graphs of functions of the form f b ± a c. can be used to sketch Eample 5 Transformations of Graphs of Eponential Functions You can review the techniques for transforming the graph of a function in Section.7. Each of the following graphs is a transformation of the graph of f 3. a. Because g 3 f, the graph of g can be obtained b shifting the graph of f one unit to the left, as shown in Figure 3.5. b. Because h 3 f, the graph of h can be obtained b shifting the graph of f downward two units, as shown in Figure 3.. c. Because k 3 f, the graph of k can be obtained b reflecting the graph of f in the -ais, as shown in Figure 3.7. d. Because j 3 f, the graph of j can be obtained b reflecting the graph of f in the -ais, as shown in Figure 3.8. g() = 3 + 3 f() = 3 f() = 3 h() = 3 FIGURE 3.5 Horizontal shift FIGURE 3. Vertical shift f() = 3 3 k() = 3 j() = 3 f() = 3 FIGURE 3.7 Reflection in -ais FIGURE 3.8 Reflection in -ais Now tr Eercise 3. Notice that the transformations in Figures 3.5, 3.7, and 3.8 keep the -ais as a horizontal asmptote, but the transformation in Figure 3. ields a new horizontal asmptote of. Also, be sure to note how the -intercept is affected b each transformation.
0 Chapter 3 Eponential and Logarithmic Functions 3 (, e) f() = e (, e ) (0, ) (, e ) FIGURE 3.9 The Natural Base e In man applications, the most convenient choice for a base is the irrational number e.78888.... This number is called the natural base. The function given b f e is called the natural eponential function. Its graph is shown in Figure 3.9. Be sure ou see that for the eponential function f e, e is the constant.78888..., whereas is the variable. Eample Evaluating the Natural Eponential Function Use a calculator to evaluate the function given b f e at each indicated value of. a. b. c. 0.5 d. 0.3 8 7 f() = e 0. 5 3 3 3 FIGURE 3.0 8 7 5 3 g() = e 0.58 3 3 FIGURE 3. Solution Function Value Graphing Calculator Kestrokes Displa a. f e e ENTER 0.353353 b. f e e ENTER 0.37879 c. f 0.5 e 0.5 e 0.5 ENTER.805 d. f 0.3 e 0.3 e 0.3 ENTER 0.7088 Now tr Eercise 33. Eample 7 Graphing Natural Eponential Functions Sketch the graph of each natural eponential function. a. f e 0. b. g e 0.58 Solution 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 3.0 and 3.. Note that the graph in Figure 3.0 is increasing, whereas the graph in Figure 3. is decreasing. 3 0 3 f 0.97.38.573.000.5 3.3.09 g.89.595 0.893 0.500 0.80 0.57 0.088 Now tr Eercise.
Section 3. Eponential Functions and Their Graphs Applications One of the most familiar eamples of eponential growth is an investment earning continuousl compounded interest. Using eponential functions, ou can develop a formula for interest compounded n times per ear and show how it leads to continuous compounding. Suppose a principal P is invested at an annual interest rate r, compounded once per ear. If the interest is added to the principal at the end of the ear, the new balance is P 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 3 P 3 P r P r r P r 3.... t P t P r t To accommodate more frequent (quarterl, monthl, or dail) compounding of interest, let n be the number of compoundings per ear and let t be the number of ears. Then the rate per compounding is r n and the account balance after t ears is A P r n nt. Amount (balance) with n compoundings per ear m 0 00,000 0,000 00,000,000,000 0,000,000 m m.5937.708389.79393.78597.78837.78809.78893 e If ou let the number of compoundings n increase without bound, the process approaches what is called continuous compounding. In the formula for n compoundings per ear, let m n r. This produces A P r n nt P r mr mrt P m mrt P m m rt. Amount with n compoundings per ear Substitute mr for n. Simplif. Propert of eponents As m increases without bound, the table at the left shows that m m e as m. From this, ou can conclude that the formula for continuous compounding is A Pe rt. Substitute e for m m.
Chapter 3 Eponential and Logarithmic Functions WARNING / CAUTION 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. Solution a. For quarterl compounding, ou have n. So, in 5 ears at 9%, the balance is A P r n nt,000 0.09 (5) $8,7.. 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,000 0.09 (5) Formula for compound interest Substitute for P, and t. $8,788.7. 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. $8,89.75. Use a calculator. Now tr Eercise 59. In Eample 8, note that continuous compounding ields more than quarterl or monthl compounding. This is tpical of the two tpes of compounding. That is, for a given principal, interest rate, and time, continuous compounding will alwas ield a larger balance than compounding n times per ear. r, n, r, n,
Section 3. Eponential Functions and Their Graphs 3 Eample 9 Radioactive Deca The half-life of radioactive radium Ra is about 599 ears. That is, for a given amount of radium, half of the original amount will remain after 599 ears. After another 599 ears, one-quarter of the original amount will remain, and so on. Let represent the mass, in grams, of a quantit of radium. The quantit present after t ears, then, is 5 t 599. a. What is the initial mass (when t 0)? b. How much of the initial mass is present after 500 ears? Algebraic Solution a. 5 t 599 5 0 599 Write original equation. Substitute 0 for t. 5 Simplif. So, the initial mass is 5 grams. b. 5 t 599 5 500 599 Write original equation. Substitute 500 for t. 5.53 8. Simplif. Use a calculator. So, about 8. grams is present after 500 ears. Now tr Eercise 73. Graphical Solution Use a graphing utilit to graph 5 t 599. a. Use the value feature or the zoom and trace features of the graphing utilit to determine that when 0, the value of is 5, as shown in Figure 3.. So, the initial mass is 5 grams. b. Use the value feature or the zoom and trace features of the graphing utilit to determine that when 500, the value of is about 8., as shown in Figure 3.3. So, about 8. grams is present after 500 ears. 30 0 5000 0 0 0 FIGURE 3. FIGURE 3.3 30 5000 CLASSROOM DISCUSSION Identifing Eponential Functions Which of the following functions generated the two tables below? Discuss how ou were able to decide. What do these functions have in common? Are an of them the same? If so, eplain wh. a. f b. f c. f 3 3 8 3 d. f e. f f. f 8 5 7 7 0 3 g 7.5 8 9 5 0 h 3 8 Create two different eponential functions of the forms a b and c d with -intercepts of 0, 3.
Chapter 3 Eponential and Logarithmic Functions 3. EXERCISES See www.calcchat.com for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. Polnomial and rational functions are eamples of functions.. Eponential and logarithmic functions are eamples of nonalgebraic functions, also called functions. 3. You can use the Propert to solve simple eponential equations.. The eponential function given b f e is called the function, and the base e is called the base. 5. To find the amount A in an account after t ears with principal P and an annual interest rate r compounded n times per ear, ou can use the formula.. To find the amount A in an account after t ears with principal P and an annual interest rate r compounded continuousl, ou can use the formula. SKILLS AND APPLICATIONS In Eercises 7, evaluate the function at the indicated value of. Round our result to three decimal places. Function Value 7. 8. 9. 0... f 0.9 f.3 f 5 f 3 5 g 5000 f 00.. 3 3 0.5 In Eercises 3, match the eponential function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) (c) (0, ) (0, ) (d) 3. f. f 5. f. f (0, ( (0, ) In Eercises 7, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. 7. f 8. f 9. f 0. f. f. f 3 3 In Eercises 3 8, use the graph of f to describe the transformation that ields the graph of g. 3.. 5.. 7. 8. f 3, f, f, f 0, f 7, f 0.3, g 3 g 3 g 3 3 g 0 g 7 g 0.3 5 In Eercises 9 3, use a graphing utilit to graph the eponential function. 9. 30. 3 3. 3 3. In Eercises 33 38, evaluate the function at the indicated value of. Round our result to three decimal places. Function Value 33. h e 3 3. f e 3. 35. f e 5 0 3. f.5e 0 37. f 5000e 0.0 38. f 50e 0.05 0
Section 3. Eponential Functions and Their Graphs 5 In Eercises 39, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. 39. f e 0. f e. f 3e. f e 0.5 3. f e. f e 5 In Eercises 5 50, use a graphing utilit to graph the eponential function. 5..08 5..08 5 7. s t e 0.t 8. s t 3e 0.t 9. g e 50. h e In Eercises 5 58, use the One-to-One Propert to solve the equation for. 5. 3 7 5. 3 53. 3 5. 5 5 55. e 3 e 3 5. e e 57. e 3 e 58. e e 5 COMPOUND INTEREST In Eercises 59, 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 35 Continuous A 59. P $500, r %, t 0 ears 0. P $500, r 3.5%, t 0 ears. P $500, r %, t 0 ears. P $000, r %, t 0 ears COMPOUND INTEREST In Eercises 3, complete the table to determine the balance A for $,000 invested at rate r for t ears, compounded continuousl. t 0 0 30 0 50 A 3. r %. r % 5. r.5%. r 3.5% 7. TRUST FUND On the da of a child s birth, a deposit of $30,000 is made in a trust fund that pas 5% interest, compounded continuousl. Determine the balance in this account on the child s 5th birthda. 8. 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? 9. 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 $3.95. Estimate the price 0 ears from now. 70. COMPUTER VIRUS The number V of computers infected b a computer virus increases according to the model V t 00e.05t, where t is the time in hours. Find the number of computers infected after (a) hour, (b).5 hours, and (c) hours. 7. POPULATION GROWTH The projected populations of California for the ears 05 through 030 can be modeled b P 3.9e 0.0098t, where P is the population (in millions) and t is the time (in ears), with t 5 corresponding to 05. (Source: U.S. Census Bureau) (a) Use a graphing utilit to graph the function for the ears 05 through 030. (b) Use the table feature of a graphing utilit to create a table of values for the same time period as in part (a). (c) According to the model, when will the population of California eceed 50 million? 7. POPULATION The populations P (in millions) of Ital from 990 through 008 can be approimated b the model P 5.8e 0.005t, where t represents the ear, with t 0 corresponding to 990. (Source: U.S. Census Bureau, International Data Base) (a) According to the model, is the population of Ital increasing or decreasing? Eplain. (b) Find the populations of Ital in 000 and 008. (c) Use the model to predict the populations of Ital in 05 and 00. 73. RADIOACTIVE DECAY Let Q represent a mass of radioactive plutonium 39 Pu (in grams), whose halflife is,00 ears. The quantit of plutonium present after t ears is Q t,00. (a) Determine the initial quantit (when t 0). (b) Determine the quantit present after 75,000 ears. (c) Use a graphing utilit to graph the function over the interval t 0 to t 50,000.
Chapter 3 Eponential and Logarithmic Functions 7. RADIOACTIVE DECAY Let Q represent a mass of carbon C (in grams), whose half-life is 575 ears. The quantit of carbon present after t ears is (a) Determine the initial quantit (when t 0). (b) Determine the quantit present after 000 ears. (c) Sketch the graph of this function over the interval t 0 to t 0,000. 75. DEPRECIATION After t ears, the value of a wheelchair conversion van that originall cost $30,500 7 depreciates so that each ear it is worth 8 of its value for the previous ear. (a) Find a model for V t, the value of the van after t ears. (b) Determine the value of the van ears after it was purchased. 7. DRUG CONCENTRATION Immediatel following an injection, the concentration of a drug in the bloodstream is 300 milligrams per milliliter. After t hours, the concentration is 75% of the level of the previous hour. (a) Find a model for C t, the concentration of the drug after t hours. (b) Determine the concentration of the drug after 8 hours. EXPLORATION TRUE OR FALSE? In Eercises 77 and 78, determine whether the statement is true or false. Justif our answer. 77. The line is an asmptote for the graph of f 0. 78. Q 0 t 575. e 7,80 99,990 THINK ABOUT IT In Eercises 79 8, use properties of eponents to determine which functions (if an) are the same. 79. f 3 80. f g 3 9 g h 9 3 h 8. f 8. f e 3 g g e 3 h h e 3 83. Graph the functions given b 3 and and use the graphs to solve each inequalit. (a) < 3 (b) > 3 8. Use a graphing utilit to graph each function. Use the graph to find where the function is increasing and decreasing, and approimate an relative maimum or minimum values. (a) f e (b) g 3 85. GRAPHICAL ANALYSIS Use a graphing utilit to graph and e in the same viewing window. Using the trace feature, eplain what happens to the graph of as increases. 8. GRAPHICAL ANALYSIS Use a graphing utilit to graph f 0.5 and in the same viewing window. What is the relationship between f and g as increases and decreases without bound? 87. GRAPHICAL ANALYSIS Use a graphing utilit to graph each pair of functions in the same viewing window. Describe an similarities and differences in the graphs. (a) (b) 3, 3, 88. THINK ABOUT IT Which functions are eponential? (a) 3 (b) 3 (c) 3 (d) 89. COMPOUND INTEREST Use the formula A P r n nt g e 0.5 to calculate the balance of an account when P $3000, r %, and t 0 ears, and compounding is done (a) b the da, (b) b the hour, (c) b the minute, and (d) b the second. Does increasing the number of compoundings per ear result in unlimited growth of the balance of the account? Eplain. 90. CAPSTONE The figure shows the graphs of, e, 0,, e, and 0. Match each function with its graph. [The graphs are labeled (a) through (f).] Eplain our reasoning. b a c 0 8 PROJECT: POPULATION PER SQUARE MILE To work an etended application analzing the population per square mile of the United States, visit this tet s website at academic.cengage.com. (Data Source: U.S. Census Bureau) d e f