Exponential and Logarithmic Functions

Transcription

1 Because of permissions issues, some material (e.g., photographs) has been removed from this chapter, though reference to it ma occur in the tet. The omitted content was intentionall deleted and is not needed to meet the Universit's requirements for this course. Eponential and Logarithmic Functions CHAPTER Can I put aside $5,000 when I m 0 and wind up sitting on half a million dollars b m earl fifties? Will population growth lead to a future without comfort or individual choice? Wh did I feel I was walking too slowl on m visit to New York Cit? Are Californians at greater risk from drunk drivers than from earthquakes? What is the difference between earthquakes measuring 6 and 7 on the Richter scale? And what can possibl be causing merchants at our local shopping mall to grin from ear to ear as the watch the browsers? The functions that ou will be learning about in this chapter will provide ou with the mathematics for answering these questions. You will see how these remarkable functions enable us to predict the future and rediscover the past. You ll be sitting on $500,000 in Eample 8 of Section.. Here s where ou ll find the other models related to our questions: World population growth: Section.5, Eamples and 5 Population and walking speed: Section.5, Check Point, and Review Eercises, Eercise 7 Alcohol and risk of a car accident: Section., Eample 7 Earthquake intensit: Section., Eample 9. We open the chapter with those grinning merchants and the sound of ka-ching! 87

2 88 CHAPTER Eponential and Logarithmic Functions SECTION. Eponential Functions Objectives Evaluate eponential functions. Graph eponential functions. Evaluate functions with base e. Use compound interest formulas. Just browsing? Take our time. Researchers know, to the dollar, the average amount the tpical consumer spends per minute at the shopping mall. And the longer ou sta, the more ou spend. So if ou sa ou re just browsing, that s just fine with the mall merchants. Browsing is time and, as shown in Figure., time is mone. The data in Figure. can be modeled b the function f =..56, Average Amount Spent $0 $00 $60 $0 $80 $0 FIGURE. Mall Browsing Time and Average Amount Spent Time at a Shopping Mall (hours) where f is the average amount spent, in dollars, at a shopping mall after hours. Can ou see how this function is different from polnomial functions? The variable is in the eponent. Functions whose equations contain a variable in the eponent are called eponential functions. Man real-life situations, including population growth, growth of epidemics, radioactive deca, and other changes that involve rapid increase or decrease, can be described using eponential functions. $7 $7 $ $9 Source: International Council of Shopping Centers Research, 006 $00 Definition of an Eponential Function The eponential function f with base b is defined b f = b or = b, where b is a positive constant other than ( b 7 0 and b Z ) and is an real number. Here are some eamples of eponential functions: f()= g()=0 h()= ± j()=a b. Base is. Base is 0. Base is. Each of these functions has a constant base and a variable eponent. Base is.

3 SECTION. Eponential Functions 89 B contrast, the following functions are not eponential functions: F()= G()= H()=( ) J()=. Variable is the base and not the eponent. The base of an eponential function must be a positive constant other than. The base of an eponential function must be positive. Variable is both the base and the eponent. Wh is G = not classified as an eponential function? The number raised to an power is. Thus, the function G can be written as G =, which is a constant function. Wh is H = - not an eponential function? The base of an eponential function must be positive to avoid having to eclude man values of from the domain that result in nonreal numbers in the range: H()=( ) Not an eponential function Ha b=( ) = =i. All values of resulting in even roots of negative numbers produce nonreal numbers. Evaluate eponential functions. You will need a calculator to evaluate eponential epressions. Most scientific calculators have a ke. Graphing calculators have a ke. To evaluate epressions of the form b, enter the base b, press or, enter the eponent, = ENTER and finall press or. EXAMPLE Evaluating an Eponential Function The eponential function f =..56 models the average amount spent, f, in dollars, at a shopping mall after hours. What is the average amount spent, to the nearest dollar, after four hours? Solution Because we are interested in the amount spent after four hours, substitute for and evaluate the function. f =..56 This is the given function. f =..56 Substitute for. Use a scientific or graphing calculator to evaluate f. Press the following kes on our calculator to do this: Scientific calculator:. *.56 = Graphing calculator:. *.56 ENTER. The displa should be approimatel f =..56 L L 50 Thus, the average amount spent after four hours at a mall is $50. CHECK POINT Use the eponential function in Eample to find the average amount spent, to the nearest dollar, after three hours at a shopping mall. Does this rounded function value underestimate or overestimate the amount shown in Figure.? B how much?

4 80 CHAPTER Eponential and Logarithmic Functions Graph eponential functions. Graphing Eponential Functions We are familiar with epressions involving where is a rational number. For eample, b b.7 = b 7 0 = A b 7 and b.7 = b 7 00 = A0 b 7. However, note that the definition of f = b includes all real numbers for the domain. You ma wonder what b means when is an irrational number, such as b or b p. Using closer and closer approimations for A L.705B, we can think of b as the value that has the successivel closer approimations b.7, b.7, b.7, b.705, Á. In this wa, we can graph the eponential function with no holes, or points of discontinuit, at the irrational domain values. EXAMPLE Graph: f =. Graphing an Eponential Function Solution We begin b setting up a table of coordinates. - f () f- = - = f- = - = f- = - = f0 = 0 = Range: (0, ) 6 5 f() = f = = f = = f = = 8 Horizontal asmptote: = 0 Domain: (, ) FIGURE. The graph of f () = We plot these points, connecting them with a continuous curve. Figure. shows the graph of f =. Observe that the graph approaches, but never touches, the negative portion of the -ais. Thus, the -ais, or = 0, is a horizontal asmptote. The range is the set of all positive real numbers. Although we used integers for in our table of coordinates, ou can use a calculator to find additional points. For eample, f0. = 0. L. and f0.95 = 0.95 L.9. The points (0.,.) and (0.95,.9) approimatel fit the graph. CHECK POINT Graph: f =.

5 SECTION. Eponential Functions 8 EXAMPLE Graphing an Eponential Function Graph: g = a b. Solution We begin b setting up a table of coordinates. We compute the function values b noting that g = a b = - = -. g () a b or 8 - g- = -- = = 8 - g- = -- = = - g- = -- = = 0 g0 = -0 = g = - = = g = - = = g = - = = 8 Range: (0, ) Horizontal asmptote: = Domain: (, ) g() = ( ) FIGURE. The graph of g () = a b We plot these points, connecting them with a continuous curve. Figure. shows the graph of g = A B. This time the graph approaches, but never touches, the positive portion of the -ais. Once again, the -ais, or = 0, is a horizontal asmptote. The range consists of all positive real numbers. Do ou notice a relationship between the graphs of f = and in Figures. and.? The graph of the graph of f = about the -ais. g = A B g = A B is a mirror image, or reflection, of f = A B. CHECK POINT Graph: Note that f = A B = - = -. Four eponential functions have been graphed in Figure.. Compare the black and green graphs, where b 7, to those in blue and red, where b 6. When b 7, the value of increases as the value of increases. When b 6, the value of decreases as the value of increases. Notice that all four graphs pass through (0, ). These graphs illustrate general characteristics of eponential functions, listed in the bo on the net page. FIGURE. Graphs of four eponential functions = ( ) = ( ) = 7 Horizontal asmptote: = 0 =

6 8 CHAPTER Eponential and Logarithmic Functions Characteristics of Eponential Functions of the Form f() b. The domain of f = b consists of all real numbers: - q, q. The range of f = b consists of all positive real numbers: 0, q.. The graphs of all eponential functions of the form f = b pass through the point (0, ) because f0 = b 0 = b Z 0. The -intercept is.. If b 7, f = b has a graph that goes up to the right and is an increasing function. The greater the value of b, the steeper the increase.. If 0 6 b 6, f = b has a graph that goes down to the right and is a decreasing function. The smaller the value of b, the steeper the decrease. 5. The graph of f = b approaches, but does not touch, the -ais. The -ais, or = 0, is a horizontal asmptote. f() = b 0 < b < (0, ) Horizontal asmptote: = 0 f() = b b > EXAMPLE Graphing Eponential Functions Graph f = and g = + in the same rectangular coordinate sstem. How is the graph of g related to the graph of f? Solution We begin b setting up a table showing some of the coordinates for f and g, selecting integers from - to for. Notice that + is the eponent for g = f () f- = - = 9 f- = - = f0 = 0 = f = = g () g- = - + = - = g- = - + = 0 = g0 = 0 + = = g = + = = 9 f = = 9 g = + = = f() = g() = + (, ) (0, ) Horizontal asmptote: = 0 FIGURE.5 We plot the points for each function and connect them with a smooth curve. Because of the scale on the -ais, the points on each function corresponding to = are not shown. Figure.5 shows the graphs of f = and g = +. The graph of g is the graph of f shifted one unit to the left. CHECK POINT Graph f = and g = - in the same rectangular coordinate sstem. Select integers from - to for. How is the graph of g related to the graph of f?

7 SECTION. Eponential Functions 8 EXAMPLE 5 Graphing Eponential Functions Graph f = and g = - in the same rectangular coordinate sstem. How is the graph of g related to the graph of f? Solution We begin b setting up a table showing some of the coordinates for f and g, selecting integers from - to for f () f- = - = f- = - = f0 = 0 = f = = g () g- = - - = - = - g- = - - = - = - g0 = 0 - = - = - g = - = - = - f = = g = - = - = 5 f() = g() = Horizontal asmptote: = FIGURE.6 We plot the points for each function and connect them with a smooth curve. Figure.6 shows the graphs of f = and g = -. The graph of g is the graph of f shifted down three units. As a result, = - is the horizontal asmptote for g. CHECK POINT 5 Graph f = and g = + in the same rectangular coordinate sstem. Select integers from - to for. How is the graph of g related to the graph of f? Evaluate functions with base e. Using Technolog Graphic Connections As n increases, the graph of = A + n B n approaches the graph of = e. ( ) = + n n = e [0, 5, ] b [0,, ] n The Natural Base e An irrational number, smbolized b the letter e, appears as the base in man applied eponential functions. The number e is defined as the value that approaches A + nb n as n gets larger and larger. Table. shows values of A + nb n for increasingl large values of n. As n increases, the approimate value of e to nine decimal places is e L The irrational number e, approimatel.7, is called the natural base. The function f = e is called the natural eponential function. Table. a n n n b , , ,000, ,000,000, As n takes on increasingl large values, the epression ( + n) n approaches e.

8 8 CHAPTER Eponential and Logarithmic Functions Use a scientific or graphing calculator with an e ke to evaluate e to various powers. For eample, to find e, press the following kes on most calculators: Scientific calculator: e Graphing calculator: e ENTER. The displa should be approimatel e L 7.89 The number e lies between and. Because = and = 9, it makes sense that e, approimatel 7.89, lies between and 9. Because 6 e 6, the graph of = e is between the graphs of = and =, shown in Figure.7. = (, ) (0, ) = Horizontal asmptote: = 0 (, ) = e (, e) FIGURE.7 Graphs of three eponential functions EXAMPLE 6 Gra Wolf Population Insatiable killer. That s the reputation the gra wolf acquired in the United States in the nineteenth and earl twentieth centuries. Although the label was undeserved, an estimated two million wolves were shot, trapped, or poisoned. B 960, the population was reduced to 800 wolves. Figure.8 shows the rebounding population in two recover areas after the gra wolf was declared an endangered species and received federal protection. Gra Wolf Population in Two Recover Areas for Selected Years Northern Rock Mountains Western Great Lakes Year Year FIGURE.8 Source: U.S. Fish and Wildlife Service The eponential function f =.6e 0.7 models the gra wolf population of the Northern Rock Mountains, f, ears after 978. If the wolf is not removed from the endangered species list and trends shown in Figure.8 continue, project its population in the recover area in 00.

9 Solution Because 00 is ears after 978, we substitute for in the given function. f =.6e 0.7 f =.6e 0.7 Perform this computation on our calculator. The displa should be approimatel.97. Thus, SECTION. Eponential Functions 85 This is the given function. Substitute for. Scientific calculator:.6 *.7 * e = Graphing calculator:.6 * e.7 * ENTER f =.6e 0.7 L. This parenthesis is given on some calculators. This indicates that the gra wolf population of the Northern Rock Mountains in the ear 00 is projected to be. CHECK POINT 6 The eponential function f = 066e 0.0 models the gra wolf population of the Western Great Lakes, f, ears after 978. If trends shown in Figure.8 continue, project the gra wolf s population in the recover area in 0. Use compound interest formulas. Compound Interest In Chapter, we saw that the amount of mone, A, that a principal, P, will be worth after t ears at interest rate r, compounded annuall, is given b the formula A = P + r t. Most savings institutions have plans in which interest is paid more than once a ear. If compound interest is paid twice a ear, the compounding period is si months. We sa that the interest is compounded semiannuall. When compound interest is paid four times a ear, the compounding period is three months and the interest is said to be compounded quarterl. Some plans allow for monthl compounding or dail compounding. In general, when compound interest is paid n times a ear, we sa that there are n compounding periods per ear. The formula A = P + r t can be adjusted to take into account the number of compounding periods in a ear. If there are n compounding periods per ear, the formula becomes A = Pa + r n b nt. Some banks use continuous compounding, where the number of compounding periods increases infinitel (compounding interest ever trillionth of a second, ever quadrillionth of a second, etc.). As n, the number of compounding periods in a ear, increases without bound, the epression a + n approaches e. As a result, the formula n b for continuous compounding is A = Pe rt. Although continuous compounding sounds terrific, it ields onl a fraction of a percent more interest over a ear than dail compounding. 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 compoundings per ear: A = Pa + r nt n n b. For continuous compounding: A = Pe rt.

10 86 CHAPTER Eponential and Logarithmic Functions EXAMPLE 7 Choosing between Investments You decide to invest $8000 for 6 ears and ou have a choice between two accounts. The first pas 7% per ear, compounded monthl. The second pas 6.85% per ear, compounded continuousl. Which is the better investment? Solution The better investment is the one with the greater balance in the account after 6 ears. Let s begin with the account with monthl compounding. We use the compound interest model with P = 8000, r = 7% = 0.07, n = (monthl compounding means compoundings per ear), and t = 6. A = Pa + r n b nt = 8000a b # 6 L,60.8 The balance in this account after 6 ears is $,60.8. For the second investment option, we use the model for continuous compounding with P = 8000, r = 6.85% = , and t = 6. A = Pe rt = 8000e L, The balance in this account after 6 ears is $,066.60, slightl less than the previous amount. Thus, the better investment is the 7% monthl compounding option. CHECK POINT 7 A sum of $0,000 is invested at an annual rate of 8%. Find the balance in the account after 5 ears subject to a. quarterl compounding and b. continuous compounding.. EXERCISE SET Practice Eercises In Eercises 0, approimate each number using a calculator. Round our answer to three decimal places e. 8. e. 9. e e In Eercises 6, set up a table of coordinates for each function. Select integers from - to, inclusive, for. Then use the table of coordinates to match the function with its graph. [The graphs are labeled a through f. ]. f =. f = -. f = -. f = - 5. f = - 6. f = - - b. c. a. d.

11 SECTION. Eponential Functions 87 e. 5. f = and g = f = and g = f = and g = # 8. f = and g = # f. In Eercises 7, graph each function b making a table of coordinates. If applicable, use a graphing utilit to confirm our hand-drawn graph. 7. f = 8. f = g = a g = a b b.. h = a h = a b b. f = 0.6. f = 0.8 In Eercises 5 8, graph functions f and g in the same rectangular coordinate sstem. Select integers from - to, inclusive, for. Then describe how the graph of g is related to the graph of f. If applicable, use a graphing utilit to confirm our handdrawn graphs. 5. f = and g = + 6. f = and g = + Use the compound interest formulas, A = Pa + r nt and n b A = Pe rt, to solve Eercises 9. Round answers to the nearest cent. 9. Find the accumulated value of an investment of $0,000 for 5 ears at an interest rate of 5.5% if the mone is a. compounded semiannuall; b. compounded monthl; c. compounded continuousl. 0. Find the accumulated value of an investment of $5000 for 0 ears at an interest rate of 6.5% if the mone is a. compounded semiannuall; b. compounded monthl; c. compounded continuousl.. Suppose that ou have $,000 to invest. Which investment ields the greater return over ears: 7% compounded monthl or 6.85% compounded continuousl?. Suppose that ou have $6000 to invest. Which investment ields the greater return over ears: 8.5% compounded quarterl or 8.% compounded semiannuall? Practice PLUS In Eercises 8, use each eponential function s graph to determine the function s domain and range.. 7. f = and g = - f() = 8. f = and 9. f = and g = - g = + 0. f = and g = +. f = and g = -.. f = and. f = and g = - g = -. f = and g = - f() =

12 88 CHAPTER Eponential and Logarithmic Functions f() = + f() = + f() = ( ) + f() = ( ) + The eponential function f = models the population of India, f, in millions, ears after 97. a. Substitute 0 for and, without using a calculator, find India s population in 97. b. Substitute 7 for and use our calculator to find India s population, to the nearest million, in the ear 00 as modeled b this function. c. Find India s population, to the nearest million, in the ear 08 as predicted b this function. d. Find India s population, to the nearest million, in the ear 055 as predicted b this function. e. What appears to be happening to India s population ever 7 ears? 5. The 986 eplosion at the Chernobl nuclear power plant in the former Soviet Union sent about 000 kilograms of radioactive cesium-7 into the atmosphere. The function f = 000(0.5) 0 describes the amount, f, in kilograms, of cesium-7 remaining in Chernobl ears after 986. If even 00 kilograms of cesium-7 remain in Chernobl s atmosphere, the area is considered unsafe for human habitation. Find f80 and determine if Chernobl will be safe for human habitation b 066. The formula S = C + r t models inflation, where C = the value toda, r = the annual inflation rate, and S = the inflated value t ears from now. Use this formula to solve Eercises Round answers to the nearest dollar. 55. If the inflation rate is 6%, how much will a house now worth $65,000 be worth in 0 ears? 56. If the inflation rate is %, how much will a house now worth $50,000 be worth in 5 ears? e Use a calculator with an ke to solve Eercises The graph shows the number of words, in millions, in the U.S. federal ta code for selected ears from 955 through 005. The data can be modeled b f = and g =.87e 0.0 In Eercises 9 50, graph f and g in the same rectangular coordinate sstem. Then find the point of intersection of the two graphs. 9. f =, g = f = +, g = Graph = and = in the same rectangular coordinate sstem. 5. Graph = and = in the same rectangular coordinate sstem. Application Eercises Use a calculator with a ke or a ke to solve Eercises India is currentl one of the world s fastest-growing countries. B 00, the population of India will be larger than the population of China; b 050, nearl one-third of the world s population will live in these two countries alone. in which f and g represent the number of words, in millions, in the federal ta code ears after 955. Use these functions to solve Eercises Round answers to one decimal place. Number of Words (millions) Source: The Ta Foundation Number of Words, in Millions, in the Federal Ta Code Year

13 SECTION. Eponential Functions a. According to the linear model, how man millions of words were in the federal ta code in 005? b. According to the eponential model, how man millions of words were in the federal ta code in 005? c. Which function is a better model for the data in 005? 58. a. According to the linear model, how man millions of words were in the federal ta code in 975? b. According to the eponential model, how man millions of words were in the federal ta code in 975? c. Which function is a better model for the data in 975? 59. In college, we stud large volumes of information information that, unfortunatel, we do not often retain for ver long. The function describes the percentage of information, f, that a particular person remembers weeks after learning the information. a. Substitute 0 for and, without using a calculator, find the percentage of information remembered at the moment it is first learned. b. Substitute for and find the percentage of information that is remembered after week. c. Find the percentage of information that is remembered after weeks. d. Find the percentage of information that is remembered after one ear (5 weeks). 60. In 66, Peter Minuit persuaded the Wappinger Indians to sell him Manhattan Island for $. If the Native Americans had put the $ into a bank account paing 5% interest, how much would the investment have been worth in the ear 005 if interest were compounded a. monthl? b. continuousl? The function f = 80e f = e -0. models the percentage, f, of people ears old with some coronar heart disease. Use this function to solve Eercises 6 6. Round answers to the nearest tenth of a percent. 6. Evaluate f0 and describe what this means in practical terms. 6. Evaluate f70 and describe what this means in practical terms. 6. The function describes the number of people, Nt, who become ill with influenza t weeks after its initial outbreak in a town with 0,000 inhabitants. The horizontal asmptote in the graph indicates that there is a limit to the epidemic s growth. Number of Ill People 0,000 N(t) Nt = Number of people initiall ill with the flu a. How man people became ill with the flu when the epidemic began? (When the epidemic began, t = 0. ) b. How man people were ill b the end of the third week? c. Wh can t the spread of an epidemic simpl grow indefinitel? What does the horizontal asmptote shown in the graph indicate about the limiting size of the population that becomes ill? Writing in Mathematics 6. What is an eponential function? 65. What is the natural eponential function? 66. Use a calculator to obtain an approimate value for e to as man decimal places as the displa permits. Then use the calculator to evaluate a + for = 0, 00, 000, 0,000, b 00,000, and,000,000. Describe what happens to the epression as increases. 67. Write an eample similar to Eample 7 on page 86 in which continuous compounding at a slightl lower earl interest rate is a better investment than compounding n times per ear. 68. Describe how ou could use the graph of f = to obtain a decimal approimation for. Technolog Eercises 0, e -.5t N(t) = Time 0, e.5t 69. You have $0,000 to invest. One bank pas 5% interest compounded quarterl and the other pas.5% interest compounded monthl. a. Use the formula for compound interest to write a function for the balance in each account at an time t in ears. b. Use a graphing utilit to graph both functions in an appropriate viewing rectangle. Based on the graphs, which bank offers the better return on our mone? t

14 80 CHAPTER Eponential and Logarithmic Functions 70. a. Graph = e and = + + in the same viewing rectangle. b. Graph = e and = + + in the same + 6 viewing rectangle. c. Graph = e and = + + in the same viewing rectangle. d. Describe what ou observe in parts (a) (c).tr generalizing this observation. 77. If f =, then fa + b = fa + fb. 78. The functions f = A and g = - B have the same graph. 79. The graphs labeled (a) (d) in the figure represent =, = 5, = A and = A 5 B B,, but not necessaril in that order. Which is which? Describe the process that enables ou to make this decision. (a) (b) (c) (d) Critical Thinking Eercises Make Sense? In Eercises 7 7, determine whether each statement makes sense or does not make sense and eplain our reasoning. 7. M graph of f = # shows that the horizontal asmptote for f is =. 7. I m using a photocopier to reduce an image over and over b 50%, so the eponential function f = A B models the new image size, where is the number of reductions. 7. I m looking at data that show ogurt sales in the United States, in billions of dollars, for selected ears from 978 through 006, and a linear function appears to be a better choice than an eponential function for modeling sales during this period. 80. The hperbolic cosine and hperbolic sine functions are defined b cosh = e + e - Prove that cosh - sinh =. 0 (0, ) and sinh = e - e -. United States Yogurt Sales Sales (billions of dollars) Source: AC Nielsen Year Review Eercises ab 8. Solve for b: D = a + b. (Section 7.6, Eample 7) + 8. Subtract: (Section 7., Eample 7) 8. Solve: - = 0. (Section 6.6, Eample 6) 7. I use the natural base e when determining how much mone I d have in a bank account that earns compound interest subject to continuous compounding. In Eercises 75 78, determine whether each statement is true or false. If the statement is false, make the necessar change(s) to produce a true statement. 75. As the number of compounding periods increases on a fied investment, the amount of mone in the account over a fied interval of time will increase without bound. 76. The functions f = - and g = - have the same graph. Preview Eercises Eercises 8 86 will help ou prepare for the material covered in the net section. 8. In Section 8., we used a switch-and-solve strateg for finding a function s inverse. (See the bo on page 590.) What problem do ou encounter when using this strateg to find the inverse of f =? to what power gives 5? 5? = If f() = - 5, find f - ().