Exponential and Logarithmic Functions

Transcription

1 pr i-hr /6/06 :0 PM Page 75 CHAPTER Eponential and Logarithmic Functions W HAT WENT WRONG ON THE space shuttle Challenger? Will population growth lead to a future without comfort or individual choice? Can I put aside a small amount of mone and have millions for earl retirement? Wh did I feel I was walking too slowl on m visit to New York Cit? Wh are people in California at far 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 I hope to accomplish in weightlifting? 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 VE RECENTLY TAKEN UP weightlifting, recording the maimum number of pounds ou can lift at the end of each week. At first our weight limit increases rapidl, but now ou notice that this growth is beginning to level off. You wonder about a function that would serve as a mathematical model to predict the number of pounds ou can lift as ou continue the sport. This problem appears as Eercise 5 in Eercise Set.5 and as the group project (Eercise 64) on page

2 pr i-hr 76 /6/06 :0 PM Page 76 Chapter Eponential and Logarithmic Functions SECTION. Eponential Functions Objectives ❶ ❷ ❸ ❹ Evaluate eponential functions. Graph eponential functions. Evaluate functions with base e. Use compound interest formulas. The space shuttle Challenger eploded approimatel 7 seconds into flight on Januar 8, 986. The traged involved damage to O-rings, which were used to seal the connections between different sections of the shuttle engines. The number of O-rings damaged increases dramaticall as temperature falls. The function f = models the number of O-rings epected to fail when the temperature is F. 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. Definition of the 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()= Base is. g()=0 Base is 0. h()= ± Base is. j()=a b. Base is. Each of these functions has a constant base and a variable eponent. B contrast, the following functions are not eponential functions: F()= Variable is the base and not the eponent. G()= The base of an eponential function must be a positive constant other than. H()=( ) The base of an eponential function must be positive. J()=. 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.

3 Section. Eponential Functions 77 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()=( ) Ha b=( ) = =i. ❶ Evaluate eponential functions. Not an eponential function All values of resulting in even roots of negative numbers produce nonreal numbers. 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, and finall press = or ENTER. EXAMPLE Evaluating an Eponential Function The eponential function f = describes the number of O-rings epected to fail, f, when the temperature is F. On the morning the Challenger was launched, the temperature was F, colder than an previous eperience. Find the number of O-rings epected to fail at this temperature. Solution function. Because the temperature was F, substitute for and evaluate the f = f = This is the given function. Substitute for. Use a scientific or graphing calculator to evaluate f. Press the following kes on our calculator to do this: Scientific calculator:.49 * = Graphing calculator:.49 * ENTER. The displa should be approimatel f = L.8 L 4 Thus, four O-rings are epected to fail at a temperature of F. ❷Graph eponential functions. Point Use the function in Eample to find the number of O-rings epected to fail at a temperature of 60 F. Round to the nearest whole number. Graphing Eponential Functions We are familiar with epressions involving b, For eample, b.7 7 = b 0 = 0 b 7 and b.7 7 = b 00 = 00 b 7. where is a rational number. 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 L.705, 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 eponential functions with no holes, or points of discontinuit, at the irrational domain values.

4 78 Chapter Eponential and Logarithmic Functions EXAMPLE Graph: Solution f =. Graphing an Eponential Function We begin b setting up a table of coordinates. f f- = - = 8 f- = - = 4 f- = - = f0 = 0 = f = = f = = 4 f = = 8 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 Figure. The graph of f = 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. Range: (0, ) Horizontal asmptote: = Domain: (, ) f() = Point Graph: f =. 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 or b g- = -- = = 8 g- = -- = = 4 g- = -- = = g0 = -0 = g = - = = g = - = = 4 Range: (0, ) Horizontal asmptote: = Domain: (, ) g() = ( ) g = - = = 8 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

5 Section. Eponential Functions 79 positive portion of the -ais. Once again, the -ais, or asmptote. The range consists of all positive real numbers. = 0, is a horizontal Do ou notice a relationship between the graphs of g = A B in Figures. and.? The graph of g = A B f = reflected about the -ais: f = and is the graph of g()=a b = =f( ) Recall that the graph of = f( ) is the graph of = f() reflected about the -ais. Point Graph: f = A B. 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,. = ( ) = ( 4 ) = 7 = Figure. Graphs of four eponential functions Horizontal asmptote: = 0 These graphs illustrate the following general characteristics of eponential functions: Characteristics of Eponential Functions of the Form f b - q, q. f0 = b = b Z 0.. The domain of f = b consists of all real numbers: 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 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. 4. 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. f = b is one-to-one and has an f() = b 0 < b < f() = b b > inverse that is a function. 6. The graph of f = b (0, ) approaches, but does not touch, the -ais. The -ais, or = 0, is a horizontal asmptote. Horizontal asmptote: = 0

6 80 Chapter Eponential and Logarithmic Functions Transformations of Eponential Functions The graphs of eponential functions can be translated verticall or horizontall, reflected, stretched, or shrunk. These transformations are summarized in Table.. Table. Transformations Involving Eponential Functions In each case, c represents a positive real number. Transformation Equation Description Vertical translation g = b + c Shifts the graph of f = b g = b - c upward c units. Shifts the graph of f = b downward c units. Horizontal translation g = b + c Shifts the graph of f = b g = b - c to the left c units. Shifts the graph of f = b to the right c units. Reflection g = -b g = b - Reflects the graph of f = b about the -ais. Reflects the graph of f = b about the -ais. Vertical stretching or g = cb Verticall stretches the graph shrinking of f = b if c 7. Verticall shrinks the graph of f = b if 0 6 c 6. Horizontal stretching or g = b c Horizontall shrinks the shrinking graph of f = b if c 7 Horizontall stretches the graph of f = b if 0 6 c 6. EXAMPLE 4 Transformations Involving Eponential Functions + Use the graph of f = to obtain the graph of g =. Solution The graph of g = + is the graph of f = shifted unit to the left. Begin with a table showing some of the coordinates for f. Graph f() =. We identified three points and the horizontal asmptote. The graph g() = + with three points and the horizontal asmptote labeled 0 f() f( )= = f( )= = f(0)= 0 = f()= = f()= = Graph g() = +. Shift f unit left. Subtract from each -coordinate. 4 f() = (, ) (, ) (, a) (0, ) (, a) (0, ) g() = + Horizontal asmptote: = 0 Horizontal asmptote: = 0 Point4 Use the graph of f = to obtain the graph of g = -.

7 Section. Eponential Functions 8 If an eponential function is translated upward or downward, the horizontal asmptote is shifted b the amount of the vertical shift. EXAMPLE 5 Transformations Involving Eponential Functions Use the graph of f = to obtain the graph of g = -. Solution The graph of g = - is the graph of f = shifted down units. Begin with a table showing some of the coordinates for f. Graph f() =. We identified three points and the horizontal asmptote. The graph g() = with three points and the horizontal asmptote labeled 0 f() f( )= = f( )= = f(0)= 0 = f()= = f()= =4 4 (, ~) Horizontal asmptote: = (0, ) (, 4) Graph g() =. Shift f units down. Subtract from each -coordinate. f() = (,!) (0, ) 5 4 (, ) g() = Horizontal asmptote: = Point5 Use the graph of f = to obtain the graph of g = +. ❸Evaluate functions with base e. Technolog As n : q, the graph of = A + n B n approaches the graph of = e. = e ( ) n = + n n [0, 5, ] b [0,, ] 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 A + nb n approaches as n gets larger and larger. Table. shows the values of A + nb n for increasingl large values of n.as n : q, the approimate value of e to nine Table. 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. 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 n a n b n , , ,000, ,000,000, As n : q, a + n b n : e.

8 8 Chapter Eponential and Logarithmic Functions 4 = (, ) (0, ) = World Population (billions) Horizontal asmptote: = 0 Figure.4 Graphs of three eponential functions (, ) = e (, e) 0 The number e lies between and. Because = 4 and = 9, it makes sense that e, approimatel 7.89, lies between 4 and 9. Because 6 e 6, the graph of = e is between the graphs of = and =, shown in Figure.4. EXAMPLE 6 World Population In a report entitled Resources and Man, the U.S. National Academ of Sciences concluded that a world population of 0 billion is close to (if not above) the maimum that an intensel managed world might hope to support with some degree of comfort and individual choice. At the time the report was issued in 969, world population was approimatel.6 billion, with a growth rate of % per ear. The function f =.6e 0.0 describes world population, f, in billions, ears after 969. Use the function to find world population in the ear 00. Is there cause for alarm? Solution Because 00 is 5 ears after 969, we substitute 5 for in f =.6e 0.0 : f5 =.6e : Estimated population Perform this computation on our calculator. Scientific calculator:.6 *.0 * 5 e = Graphing calculator:.6 * e.0 * 5 ENTER The displa should be approimatel Thus, f5 =.6e 0.05 L This indicates that world population in the ear 00 will be approimatel 9.98 billion. Because this number is quite close to 0 billion, the given function suggests that there ma be cause for alarm Source: U.N. Population Division 50 World population in 004 was approimatel 6.4 billion, but the growth rate was no longer %. It had slowed down to.%. Using this current growth rate, eponential functions now predict a world population of 7.8 billion in the ear 00. Eperts think the population ma stabilize at 0 billion after 00 if the growth rate continues to decline. Point6 The function f = 6.4e 0.0 describes world population, f, in billions, ears after 004 subject to a growth rate of.% annuall. Use the function to predict world population in 050. ❹ Use compound interest formulas. Compound Interest We all want a wonderful life with fulfilling work, good health, and loving relationships. And let s be honest: Financial securit wouldn t hurt! Achieving this goal depends on understanding how mone in savings accounts grows in remarkable was as a result of compound interest. Compound interest is interest computed on our original investment as well as on an accumulated interest. Suppose a sum of mone, called the principal, P, is invested at an annual percentage rate r, in decimal form, compounded once per ear. Because the interest is added to the principal at ear s end, the accumulated value, A,is A = P + Pr = P + r.

9 Table. Section. Eponential Functions 8 The accumulated amount of mone follows this pattern of multipling the previous principal b + r for each successive ear, as indicated in Table.. Time in Years 0 4 o t Accumulated Value after Each Compounding A = P A = P + r A = P + r + r = P + r A = P + r + r = P + r A = P + r + r = P + r 4 o A = P + r t This formula gives the balance, A, that a principal, P, is worth after t ears at interest rate r, compounded once a ear. 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, in each time period the interest rate is i = r n and there are nt time periods in t ears. This results in the following formula for the balance, A, after t ears: 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.). Let s see what happens to the balance, A, as n : q. n r rt = nt nt r h r A=P a+ b =P Ca+ n b S =P a+ =Pe rt n C r h b S n rt rt n Let h = r. As h S, b definition h As n S, h S. ( + S e. h ) We see that the formula 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 n compoundings per ear: A = Pa + r nt n b. For continuous compounding: A = Pe rt.

10 84 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,60.84 The balance in this account after 6 ears is $, 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. Point7 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.4 9. e e In Eercises 8, graph each function b making a table of coordinates. If applicable, use a graphing utilit to confirm our hand-drawn graph.. f = 4. f = 5. g = A B 4. g = A 4 B 5. h = A B 6. h = A B 7. f = f = 0.8 In Eercises 9 4, the graph of an eponential function is given. Select the function for each graph from the following options: f =, g = -, h = -, F = -, G = -, H = In Eercises 5 4, begin b graphing f =. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asmptotes. Use the graphs to determine each function s domain and range. If applicable, use a graphing utilit to confirm our hand-drawn graphs. 5. g = + 6. g = + 7. g = - 8. g = + 9. h = h =

11 Section. Eponential Functions 85. g = -. g = -. g = # 4. g = # The figure shows the graph of f = e. In Eercises 5 46, use transformations of this graph to graph each function. Be sure to give equations of the asmptotes. Use the graphs to determine each function s domain and range. If applicable, use a graphing utilit to confirm our hand-drawn graphs. (, e 0.7) (, e 0.4) (0, ) Horizontal asmptote: = 0 (, e 7.9) (, e.7) f() = e 56. Suppose that ou have $6000 to invest. Which investment ields the greater return over 4 ears: 8.5% compounded quarterl or 8.% compounded semiannuall? Practice Plus In Eercises 57 58, graph f and g in the same rectangular coordinate sstem. Then find the point of intersection of the two graphs. 57. f =, g = f = +, g = Graph = and = in the same rectangular coordinate sstem. 60. Graph = and = in the same rectangular coordinate sstem. In Eercises 6 64, give the equation of each eponential function whose graph is shown g = e - 6. g = e + 6 (, 6) 7. g = e + 8. g = e - 9. h = e h = e - 4. h = -e 4. g = e 44. g = e 45. h = e h = e + In Eercises 47 5, graph functions f and g in the same rectangular coordinate sstem. Graph and give equations of all asmptotes. If applicable, use a graphing utilit to confirm our hand-drawn graphs. 47. f = and g = f = and g = f = and g = # 50. and g = # f = 5. f = A and g = A B - B + 5. f = A and g = A B - B + + h = e - nt r Use the compound interest formulas A = Pa + and n b A = Pe rt to solve Eercises Round answers to the nearest cent. 5. 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 quarterl; c. compounded monthl; d. compounded continuousl. 54. 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 quarterl; c. compounded monthl; d. compounded continuousl. 55. Suppose that ou have $,000 to invest. Which investment ields the greater return over ears: 7% compounded monthl or 6.85% compounded continuousl? (0, ) 4 (, 4) = 0 (, ~) 40 (, 6) 4 6 (0, ) 8 (, 6) = 0 (, Z) = 0 (0, ) (, e ) 4 (, e) 6 (, e ) 8 0

12 86 Chapter Eponential and Logarithmic Functions (, e ) (, e) (0, ) (, e) = A decimal approimation for p is.459. Use a calculator to find,.,.4,.4,.45,.459, and.459. Now find p. What do ou observe? 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 000. The data can be modeled b f = and g =.8e 0.04, 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 7 7. 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 040, 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. The eponential function f = models the population of India, f, in millions, ears after 974. a. Substitute 0 for and, without using a calculator, find India s population in 974. 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? 66. 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 0 f = 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. 67. If the inflation rate is 6%, how much will a house now worth $465,000 be worth in 0 ears? 68. If the inflation rate is %, how much will a house now worth $50,000 be worth in 5 ears? 69. A decimal approimation for is Use a calculator to find.7,.7,.7,.705, and Now find. What do ou observe? Number of Words (millions) Number of Words, in Millions, in the Federal Ta Code Source: The Ta Foundation Year Which function, the linear or the eponential, is a better model for the data in 000? 7. Which function, the linear or the eponential, is a better model for the data in 985? 7. In college, we stud large volumes of information information that, unfortunatel, we do not often retain for ver long. The function f = 80e 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 4 weeks. d. Find the percentage of information that is remembered after one ear (5 weeks). 74. In 66, Peter Minuit convinced the Wappinger Indians to sell him Manhattan Island for $4. If the Native Americans had put the $4 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?

13 Section. Eponential Functions 87 The bar graph shows the number of identit theft complaints to the Federal Trade Commission from 000 through 004. (The problem is much worse: The graph shows onl the complaints. According to an FCC surve, 9.9 million Americans about in 0 were victims of identit theft from spring 00 to spring 00.) Number of Complaints 70,000 40,000 0,000 80,000 50,000 0,000 90,000 60,000 0,000 Number of Identit Theft Complaints to the Federal Trade Commission, , 00 Source: Federal Trade Commission The functions f = 6, Year 5, , , e -. and g = 55, ,7.8 model the number of identit theft complaints to the FCC, f or g, ears after 000. Use these functions to solve Eercises Which function is a better model for the number of complaints in 004? 76. Which function is a better model for the number of complaints in 00? Writing in Mathematics 77. What is an eponential function? 78. What is the natural eponential function? 79. Use a calculator to evaluate a + for = 0, 00, 000, b 0,000, 00,000, and,000,000. Describe what happens to the epression as increases. 80. Describe how ou could use the graph of f = to obtain a decimal approimation for. 8. The eponential function = is one-to-one and has an inverse function.tr finding the inverse function b echanging and and solving for. Describe the difficult that ou encounter in this process. What is needed to overcome this problem? 8. In 004, world population was approimatel 6.4 billion with an annual growth rate of.%. Discuss two factors that would cause this growth rate to slow down over the net ten ears. Technolog Eercises 8. Graph = , the function for the number of O-rings epected to fail at F, in a [0, 90, 0] b [0, 0, 5] view- ing rectangle. If NASA engineers had used this function and its graph, is it likel the would have allowed the Challenger to be launched when the temperature was F? Eplain. 84. You have $0,000 to invest. One bank pas 5% interest compounded quarterl and the other pas 4.5% interest compounded monthl. a. Use the formula for compound interest to write a function for the balance in each account at an time t. 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? 85. 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. Critical Thinking Eercises 86. Which one of the following is true? a. 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. b. The functions f = - and g = - have the same graph. c. If f =, then fa + b = fa + fb. d. The functions f = A and g = - B have the same graph. 87. 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. 88. Graph f = and its inverse function in the same rectangular coordinate sstem. 89. The hperbolic cosine and hperbolic sine functions are defined b cosh = e + e - a. Show that cosh is an even function. b. Show that sinh is an odd function. c. Prove that cosh - sinh =. (a) (b) (c) (d) 0 (0, ) and sinh = e - e -.

14 pr i-hr 88 /6/06 :0 PM Page 88 Chapter Eponential and Logarithmic Functions SECTION. Logarithmic Functions Objectives ❶ ❷ ❸ ❹ ❺ ❻ ❼ ❽ Change from logarithmic to eponential form. Change from eponential to logarithmic form. Evaluate logarithms. Use basic logarithmic properties. Graph logarithmic functions. Find the domain of a logarithmic function. Use common logarithms. Use natural logarithms. The earthquake that ripped through northern California on October 7, 989, measured 7. on the Richter scale, killed more than 60 people, and injured more than 400. Shown here is San Francisco s Marina district, where shock waves tossed houses off their foundations and into the street. A higher measure on the Richter scale is more devastating than it seems because for each increase in one unit on the scale, there is a tenfold increase in the intensit of an earthquake. In this section, our focus is on the inverse of the eponential function, called the logarithmic function. The logarithmic function will help ou to understand diverse phenomena, including earthquake intensit, human memor, and the pace of life in large cities. The Definition of Logarithmic Functions Stud Tip The inverse of = b is = b. Logarithms give us a wa to epress the inverse function = b for in terms of. Refer to the table of contents and the section titled Inverse Functions in case ou need to review this topic. Here s a summar of what ou should alread know about functions and their inverses.. Onl one-to-one functions have inverses that are functions. A function, f, has an inverse function, f -, if there is no horizontal line that intersects the graph of f at more than one point.. If a function is one-to-one, its inverse function can be found b interchanging and in the function s equation and solving for.. If fa = b, then f -b = a. The domain of f is the range of f -. The range of f is the domain of f ff - = and f -f =. 5. The graph of f - is the reflection of the graph of f about the line =. No horizontal line can be drawn that intersects the graph of an eponential function at more than one point. This means that the eponential function is one-to-one and has an inverse. The inverse function of the eponential function with base b is called the logarithmic function with base b. Definition of the Logarithmic Function For 7 0 and b 7 0, b Z, = log b is equivalent to b =. The function f = log b is the logarithmic function with base b. The equations = log b and b = are different was of epressing the same thing. The first equation is in logarithmic form and the second equivalent equation is in eponential form. Notice that a logarithm,, is an eponent. You should learn the location of the base and eponent in each form. Location of Base and Eponent in Eponential and Logarithmic Forms Eponent Eponent Logarithmic Form: =log b Eponential Form: b= Base Base Stud Tip To change from logarithmic form to the more familiar eponential form, use this pattern: =log b means b=.

15 ❶ Change from logarithmic to eponential form. Section. Logarithmic Functions 89 EXAMPLE Changing from Logarithmic to Eponential Form Write each equation in its equivalent eponential form: a. = log 5 b. = log b 64 c. log 7 =. Solution We use the fact that = log means b b =. a. =log 5 means 5 =. b. =log b 64 means b =64. Logarithms are eponents. Logarithms are eponents. c. log or means 7 = = log 7 = 7. ❷ Change from eponential to logarithmic form. Point EXAMPLE Write each equation in its equivalent eponential form: a. = log 7 b. = log b 5 c. log 4 6 =. Changing from Eponential to Logarithmic Form Write each equation in its equivalent logarithmic form: a. = b. b = 8 c. e = 9. Solution We use the fact that b = means = log b. a. = means =log. b. b =8 means =log b 8. Eponents are logarithms. Eponents are logarithms. c. e = 9 means = log e 9. ❸Evaluate logarithms. Point Write each equation in its equivalent logarithmic form: a. 5 = b. b = 7 c. e =. Remembering that logarithms are eponents makes it possible to evaluate some logarithms b inspection. The logarithm of with base b, log b, is the eponent to which b must be raised to get. For eample, suppose we want to evaluate log We ask, to what power gives? Because 5. =, log = 5. EXAMPLE Evaluating Logarithms Evaluate: a. log 6 b. log 9 c. log 5 5. Solution Logarithmic Question Needed Logarithmic Epression Epression for Evaluation Evaluated a. log 6 to what power gives 6? log because 4 6 = 4 = 6. b. log 9 to what power gives 9? log because 9 = = 9. c. log to what power log because = = 5 = 5. gives 5? Point Evaluate: a. log 0 00 b. log c. log 6 6.

16 90 Chapter Eponential and Logarithmic Functions ❹ Use basic logarithmic properties. Basic Logarithmic Properties Because logarithms are eponents, the have properties that can be verified using properties of eponents. Basic Logarithmic Properties Involving One. log b b = because is the eponent to which b must be raised to obtain b. b = b. log b = 0 because 0 is the eponent to which b must be raised to obtain. b 0 = EXAMPLE 4 Evaluate: a. log 7 7 b. log 5. Solution Using Properties of Logarithms a. Because log b b=, we conclude log 7 7=. b. Because log b =0, we conclude log 5 =0. This means that 7 = 7. This means that 5 0 =. Point4 Evaluate: a. log 9 9 b. log 8. The inverse of the eponential function is the logarithmic function. Thus, if f = b, then f - = log b. We have seen how inverse functions undo one another. In particular, ff - = and f - f =. Appling these relationships to eponential and logarithmic functions, we obtain the following inverse properties of logarithms: Inverse Properties of Logarithms For b 7 0 and b Z, log b b = b log b = The logarithm with base b of b raised to a power equals that power. b raised to the logarithm with base b of a number equals that number. Stud Tip The voice balloons should help ou see the undoing that takes place between the eponential and logarithmic functions in the inverse properties. Start with. End with. log b b = is changed b the eponential function. Start with. End with. b log b = is changed b the logarithmic function. The change is undone b the inverse logarithmic function. The change is undone b the inverse eponential function.

17 Section. Logarithmic Functions 9 EXAMPLE 5 Using Inverse Properties of Logarithms Evaluate: a. log b. 6 log Solution a. Because log we conclude log b b =, = 5. b. Because b log b =, we conclude 6 log 6 9 = 9. ❺Graph logarithmic functions. Point5 Evaluate: a. log b. log Graphs of Logarithmic Functions How do we graph logarithmic functions? We use the fact that a logarithmic function is the inverse of an eponential function. This means that the logarithmic function reverses the coordinates of the eponential function. It also means that the graph of the logarithmic function is a reflection of the graph of the eponential function about the line = f() = = g() = log Figure.5 The graphs of f = and its inverse function EXAMPLE 6 Graphs of Eponential and Logarithmic Functions Graph f = and g = log in the same rectangular coordinate sstem. Solution We first set up a table of coordinates for f =. Reversing these coordinates gives the coordinates for the inverse function g = log. 0 f() 4 8 g() log 4 Reverse coordinates. We now plot the ordered pairs from each table, connecting them with smooth curves. Figure.5 shows the graphs of f = and its inverse function g = log. The graph of the inverse can also be drawn b reflecting the graph of f = about the line = Stud Tip You can obtain a partial table of coordinates for g = log without having to obtain and reverse coordinates for f =. Because g = log means g =, we begin with values for g and compute corresponding values for : Use = g() to compute. For eample, if g() =, = = = 4. Start with values for g(). g() log Point6 Graph f = and g = log in the same rectangular coordinate sstem.

18 9 Chapter Eponential and Logarithmic Functions Figure.6 illustrates the relationship between the graph of an eponential function, shown in blue, and its inverse, a logarithmic function, shown in red, for bases greater than and for bases between 0 and. Also shown and labeled are the eponential function s horizontal asmptote = 0 and the logarithmic function s vertical asmptote = 0. = = (0, ) Horizontal asmptote: = 0 (, 0) f () = log b f () = b f () = b (0, ) Horizontal asmptote: = 0 (, 0) f () = log b Figure.6 Graphs of eponential and logarithmic functions Vertical asmptote: = 0 Vertical asmptote: b > = 0 0 < b < Discover Verif each of the four characteristics in the bo for the red graphs in Figure.6. Characteristics of the Graphs of Logarithmic Functions of the Form f log b The -intercept is. There is no -intercept. The -ais, or = 0, is a vertical asmptote.as : 0 +, log b : - q or q. If b 7, the function is increasing. If 0 6 b 6, the function is decreasing. The graph is smooth and continuous. It has no sharp corners or gaps. The graphs of logarithmic functions can be translated verticall or horizontall, reflected, stretched, or shrunk. These transformations are summarized in Table.4. Table.4 Transformations Involving Logarithmic Functions In each case, c represents a positive real number. Transformation Equation Description Vertical translation Horizontal translation Reflection Vertical stretching or shrinking Horizontal stretching or shrinking g = log b + c g = log b - c g = log b + c g = log b - c g = -log b g = log b - g = c log b g = log b c Shifts the graph of upward c units. Shifts the graph of downward c units. Shifts the graph of f = log b to the left c units. Vertical asmptote: = -c Shifts the graph of f = log b to the right c units. Vertical asmptote: = c Reflects the graph of about the -ais. Reflects the graph of about the -ais. f = log b f = log b f = log b f = log b Verticall stretches the graph of f = log b if c 7. Verticall shrinks the graph of f = log b if 0 6 c 6. Horizontall shrinks the graph of f = log b if c 7. Horizontall stretches the graph of f = log b if 0 6 c 6.

19 Section. Logarithmic Functions 9 Vertical asmptote: = 0 = f() = log g() = log ( ) Figure.7 Shifting f = log one unit to the right For eample, Figure.7 illustrates that the graph of g = log - is the graph of f = log shifted one unit to the right. If a logarithmic function is translated to the left or to the right, both the -intercept and the vertical asmptote are shifted b the amount of the horizontal shift. In Figure.7, the -intercept of f is. Because g is shifted one unit to the right, its -intercept is. Also observe that the vertical asmptote for f, the -ais, or = 0, is shifted one unit to the right for the vertical asmptote for g. Thus, = is the vertical asmptote for g. Here are some other eamples of transformations of graphs of logarithmic functions: The graph of g = + log 4 is the graph of f = log 4 shifted up three units, shown in Figure.8. The graph of h = -log is the graph of f = log reflected about the -ais, shown in Figure.9. The graph of r = log - is the graph of f = log reflected about the -ais, shown in Figure Vertical asmptote: = 0 g() = + log 4 f() = log Vertical asmptote: = 0 f() = log h() = log 5 4 r() = log ( ) Vertical asmptote: = f() = log Figure.8 Shifting verticall up three units Figure.9 Reflection about the -ais Figure.0 Reflection about the -ais ❻ Find the domain of a logarithmic function. = f() = log 4 ( + ) The Domain of a Logarithmic Function In Section., we learned that the domain of an eponential function of the form f = b includes all real numbers and its range is the set of positive real numbers. Because the logarithmic function reverses the domain and the range of the eponential function, the domain of a logarithmic function of the form f log b is the set of all positive real numbers.thus, log 8 is defined because the value of in the logarithmic epression, 8, is greater than zero and therefore is included in the domain of the logarithmic function f = log. However, log 0 and log -8 are not defined because 0 and -8 are not positive real numbers and therefore are ecluded from the domain of the logarithmic function f = log. In general, the domain of f = log b g consists of all for which g>0. EXAMPLE 7 Finding the Domain of a Logarithmic Function Find the domain of f = log 4 +. Solution The domain of f consists of all for which Solving this inequalit for, we obtain 7 -. Thus, the domain of f is -, q. This is illustrated in Figure.. The vertical asmptote is = - and all points on the graph of f have -coordinates that are greater than -. Figure. The domain of g = log 4 + is -, q. Point7 Find the domain of f = log 4-5.

20 94 Chapter Eponential and Logarithmic Functions ❼Use common logarithms. Common Logarithms The logarithmic function with base 0 is called the common logarithmic function. The function f = log 0 is usuall epressed as f = log. A calculator with a ke can be used to evaluate common logarithms. Here are some eamples: LOG Most Scientific Most Graphing Displa (or Logarithm Calculator Kestrokes Calculator Kestrokes Approimate Displa) log log log 5 log log- LOG 5, LOG LOG ENTER LOG 5, ENTER LOG, LOG = LOG 5, LOG ENTER.9 +> - LOG LOG - ENTER ERROR Some graphing calculators displa an open parenthesis when the LOG ke is pressed. In this case, remember to close the set of parentheses after entering the function s domain value: LOG 5 ) LOG ) ENTER. The error message given b man calculators for log- is a reminder that the domain of the common logarithmic function, f = log, is the set of positive real numbers. In general, the domain of f = log g consists of all for which g 7 0. Man real-life phenomena start with rapid growth and then the growth begins to level off. This tpe of behavior can be modeled b logarithmic functions. EXAMPLE 8 Modeling Height of Children The percentage of adult height attained b a bo who is ears old can be modeled b f = log +, where represents the bo s age and f represents the percentage of his adult height. Approimatel what percentage of his adult height has a bo attained at age eight? Solution We substitute the bo s age, 8, for and evaluate the function. f = log + f8 = log8 + = log 9 L 76 This is the given function. Substitute 8 for. Graphing calculator kestrokes: LOG ENTER Thus, an 8-ear-old bo has attained approimatel 76% of his adult height. Point8 Use the function in Eample 8 to answer this question: Approimatel what percentage of his adult height has a bo attained at age ten? The basic properties of logarithms that were listed earlier in this section can be applied to common logarithms. Properties of Common Logarithms General Common Logarithm Properties Properties. log b = 0. log = 0. log b b =. log 0 =. log b b = 4. b log b = Inverse properties. log 0 = 4. 0 log =

21 Section. Logarithmic Functions 95 The propert log 0 = can be used to evaluate common logarithms involving powers of 0. For eample, log 00 = log 0 =, log 000 = log 0 =, and log 0 7. = 7.. EXAMPLE 9 Earthquake Intensit The magnitude, R, on the Richter scale of an earthquake of intensit I is given b where I 0 is the intensit of a barel felt zero-level earthquake. The earthquake that destroed San Francisco in 906 was 0 8. times as intense as a zero-level earthquake. What was its magnitude on the Richter scale? Solution Because the earthquake was 0 8. times as intense as a zero-level earthquake, the intensit, I, is 0 8. I 0. I R = log I I 0 R = log I 0 = log 0 8. = 8. I R = log, I 0 This is the formula for magnitude on the Richter scale. Substitute 0 8. I 0 for I. Simplif. Use the propert log 0. San Francisco s 906 earthquake registered 8. on the Richter scale. ❽Use natural logarithms. [ 0, 0, ] b [ 0, 0, ] Figure. The domain of f = ln - is - q,. Point9 Use the formula in Eample 9 to solve this problem. If an earthquake is 0,000 times as intense as a zero-level quake I = 0,000I 0, what is its magnitude on the Richter scale? Natural Logarithms The logarithmic function with base e is called the natural logarithmic function.the function f = log e is usuall epressed as f = ln, read el en of. A calculator with an LN ke can be used to evaluate natural logarithms. Kestrokes are identical to those shown for common logarithmic evaluations on page 94. Like the domain of all logarithmic functions, the domain of the natural logarithmic function f = ln is the set of all positive real numbers. Thus, the domain of f = ln g consists of all for which g 7 0. EXAMPLE 0 Finding Domains of Natural Logarithmic Functions Find the domain of each function: a. f = ln - b. h = ln -. Solution a. The domain of f consists of all for which Solving this inequalit for, we obtain 6. Thus, the domain of f is 5ƒ 6 6 or - q,. This is verified b the graph in Figure.. b. The domain of h consists of all for which It follows that the domain of h is all real numbers ecept.thus, the domain of h is 5ƒ Z 6 or - q,, q. This is shown b the graph in Figure.. To make it more obvious that is ecluded from the domain, we used a format. DOT [ 0, 0, ] b [ 0, 0, ] Figure. is ecluded from the domain of h = ln -. Point 0 Find the domain of each function: a. f = ln4 - b. h = ln.

22 96 Chapter Eponential and Logarithmic Functions The basic properties of logarithms that were listed earlier in this section can be applied to natural logarithms. Properties of Natural Logarithms General Natural Logarithm Properties Properties. log b = 0. ln = 0. log b b =. ln e =. log b b = 4. b logb = Inverse properties. ln e = 4. e ln = Eamine the inverse properties, ln e = and e ln =. Can ou see how ln and e undo one another? For eample, EXAMPLE Dangerous Heat: Temperature in an Enclosed Vehicle When the outside air temperature is anwhere from 7 to 96 Fahrenheit, the temperature in an enclosed vehicle climbs b 4 in the first hour. The bar graph in Figure.4 shows the temperature increase throughout the hour. The function f =.4 ln -.6 models the temperature increase, f, in degrees Fahrenheit, after minutes. Use the function to find the temperature increase, to the nearest degree, after 50 minutes. How well does the function model the actual increase shown in Figure.4? Temperature Increase ( F) ln e =, ln e 7 = 7, e ln =, and e ln 7 = 7. Temperature Increase in an Enclosed Vehicle Minutes Figure.4 Source: Professor Jan Null, San Francisco State Universit Solution We find the temperature increase after 50 minutes b substituting 50 for and evaluating the function at 50. f =.4 ln -.6 This is the given function. f50 =.4 ln Substitute 50 for. L 4 Graphing calculator kestrokes:.4 ln 50.6 ENTER. On some calculators, a parenthesis is needed after 50. According to the function, the temperature will increase b approimatel 4 after 50 minutes. Because the increase shown in Figure.4 is 4, the function models the actual increase etremel well. Point Use the function in Eample to find the temperature increase, to the nearest degree, after 0 minutes. How well does the function model the actual increase shown in Figure.4?

23 Section. Logarithmic Functions 97 The Curious Number e You will learn more about each curiosit mentioned below when ou take calculus. The number e was named b the Swiss mathematician Leonhard Euler (707 78), who proved that it is the limit as n : q of a + n. n b = e features in Euler s remarkable relationship e ip = -, in which i = -. The first few decimal places of e are fairl eas to remember: e = Á. The best approimation of e using numbers less than 000 is also eas to remember: e L 878 L.786 Á. Isaac Newton (64 77), one of the cofounders of calculus, showed that e = + + from which we obtain e = + + an infinite sum suitable for calculation because its terms decrease so rapidl. (Note: n! (n factorial) is the product of all the consecutive integers from n down to! +! + 4! + Á,! +! + 4 4! + Á, : n! = nn - n - n - # Á # # #. ) The area of the region bounded b = the -ais, = and = t (shaded in Figure.5) is a function of t, designated b, At. Grégoire de Saint-Vincent, a Belgian Jesuit ( ), spent his entire professional life attempting to find a formula for At. With his student, he showed that At = ln t, becoming one of the first mathematicians to make use of the logarithmic function for something other than a computational device. = Figure.5 A(t) = t EXERCISE SET. Practice Eercises In Eercises 8, write each equation in its equivalent eponential form.. 4 = log 6. 6 = log 64. = log 4. = log = log b 6. = log b 7 7. log 6 6 = 8. log 5 5 = In Eercises 9 0, write each equation in its equivalent logarithmic form. 9. = = = = = 4 5. = b = b = = In Eercises 4, evaluate each epression without using a calculator.. log 4 6. log log log log 5 5 log 6 6 log = 5 5 = 8 = 00 log 9 9. log log 6 6. log. log. log log log log 7. log 4 8. log 6 9. log log log log 7 4. Graph f = 4 and g = log 4 in the same rectangular coordinate sstem. 44. Graph f = 5 and g = log 5 in the same rectangular coordinate sstem. 45. Graph f = A B and g = log in the same rectangular coordinate sstem. 46. Graph f = A 4 B and g = log in the same rectangular coordinate 4 sstem. In Eercises 47 5, the graph of a logarithmic function is given. Select the function for each graph from the following options: f = log, g = log -, h = log -, F = -log, G = log -, H = - log

24 98 Chapter Eponential and Logarithmic Functions In Eercises 5 58, begin b graphing f = log. Then use transformations of this graph to graph the given function. What is the vertical asmptote? Use the graphs to determine each function s domain and range. 5. g = log g = log h = + log 56. h = + log 57. g = log 58. g = - log The figure shows the graph of f = log. In Eercises 59 64, use transformations of this graph to graph each function. Graph and give equations of the asmptotes. Use the graphs to determine each function s domain and range. Vertical asmptote: = g = log g = log - 6. h = log - 6. h = log - 6. g = - log 64. g = - log The figure shows the graph of f = ln. In Eercises 65 74, use transformations of this graph to graph each function. Graph and give equations of the asmptotes. Use the graphs to determine each function s domain and range. Vertical asmptote: = (, 0) (Å, ) (, ln.) f() = log (5, log 5 0.7) (, ln 0.7) f() = ln (, 0) (q, ln q 0.7) (0, ) 65. g = ln g = ln h = ln 68. h 69. g = ln 70. g = ln 7. g = - ln 74. g = - ln In Eercises 75 80, find the domain of each logarithmic function. 75. f = log f = log f = log f = log7-79. f = ln f = ln - 7 In Eercises 8 00, evaluate or simplif each epression without using a calculator. 8. log log log log 0 8 log ln 88. ln e 89. ln e ln e 7 9. ln 9. ln ln 5 ln e 94. e 95. ln e ln e ln 5 ln e 98. e Practice Plus In Eercises 0 04, write each equation in its equivalent eponential form. Then solve for. 0. log - = 0. log = 0. log 04. log 64 = 4 = - In Eercises 05 08, evaluate each epression without using a calculator. 05. log log log 5 log 07. log log logln e In Eercises 09, find the domain of each logarithmic function. 09. f = ln f = ln f = loga. - 5 b log 5 Application Eercises = lna B 7. h = -ln 7. h = ln- 0 log f = loga b The percentage of adult height attained b a girl who is ears old can be modeled b f = log - 4, 0 log where represents the girl s age (from 5 to 5) and f represents the percentage of her adult height. Use the function to solve Eercises 4. Round answers to the nearest tenth of a percent.. Approimatel what percentage of her adult height has a girl attained at age? 4. Approimatel what percentage of her adult height has a girl attained at age ten? e 6 e 7

25 Section. Logarithmic Functions 99 The bar graph shows the percentage of U.S. companies that performed drug tests on emploees or job applicants in five selected ears from 998 through 00. The function f = -4.9 ln models the percentage of such companies ears after 997. Use this function to solve Eercises 5 6. Round answers to the nearest percent. Percent Percentage of U.S. Companies Performing Drug Tests Year Source: American Management Association Use the function to find the percentage of U.S. companies that performed drug tests in 00. How well does this model the actual number shown for that ear? 6. Use the function to predict the percentage of U.S. companies that will be performing drug tests in 008. The loudness level of a sound, D, in decibels, is given b the formula D = 0 log0 I, where I is the intensit of the sound, in watts per meter. Decibel levels range from 0, a barel audible sound, to 60, a sound resulting in a ruptured eardrum. Use the formula to solve Eercises The sound of a blue whale can be heard 500 miles awa, reaching an intensit of 6. * 0 6 watts per meter. Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum? 8. What is the decibel level of a normal conversation,. * 0-6 watt per meter? 9. Students in a pscholog class took a final eamination. As part of an eperiment to see how much of the course content the remembered over time, the took equivalent forms of the eam in monthl intervals thereafter. The average score for the group, ft, after t months was modeled b the function ft = 88-5 lnt +, 0 t. a. What was the average score on the original eam? b. What was the average score after months? 4 months? 6 months? 8 months? 0 months? one ear? c. Sketch the graph of f (either b hand or with a graphing utilit). Describe what the graph indicates in terms of the material retained b the students. Writing in Mathematics 0. Describe the relationship between an equation in logarithmic form and an equivalent equation in eponential form.. What question can be asked to help evaluate log 8?. Eplain wh the logarithm of with base b is 0.. Describe the following propert using words: log b b =. 4. Eplain how to use the graph of f = to obtain the graph of g = log. 5. Eplain how to find the domain of a logarithmic function. 6. Logarithmic models are well suited to phenomena in which growth is initiall rapid but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function. 7. Suppose that a girl is 4 feet 6 inches at age 0. Eplain how to use the function in Eercises 4 to determine how tall she can epect to be as an adult. Technolog Eercises In Eercises 8, graph f and g in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of f. 8. f = ln, g = ln f = ln, g = ln + f = log, g = -log f = log, g = log - +. Students in a mathematics class took a final eamination. The took equivalent forms of the eam in monthl intervals thereafter. The average score, ft, for the group after t months was modeled b the human memor function ft = 75-0 logt +, where 0 t. Use a graphing utilit to graph the function. Then determine how man months will elapse before the average score falls below 65.. In parts (a) (c), graph f and g in the same viewing rectangle. a. f = ln, g = ln + ln b. f = log5, g = log 5 + log c. f = ln, g = ln + ln d. Describe what ou observe in parts (a) (c). Generalize this observation b writing an equivalent epression for log b MN, where M 7 0 and N 7 0. e. Complete this statement: The logarithm of a product is equal to. 4. Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowl to the one that increases most rapidl. =, =, = e, = ln, =, =

26 pr i-hr 400 /6/06 :0 PM Page 400 Chapter Eponential and Logarithmic Functions 8. Without using a calculator, determine which is the greater number: log 4 60 or log 40. Critical Thinking Eercises 5. Which one of the following is true? a. log 8 log 4 = Group Eercise 8 4 b. log-00 = - c. The domain of f = log is - q, q. d. logb is the eponent to which b must be raised to obtain. 6. Without using a calculator, find the eact value of log 8 - logp log 8 - log Without using a calculator, find the eact value of log4log log This group eercise involves eploring the wa we grow. Group members should create a graph for the function that models the percentage of adult height attained b a bo who is ears old, f = log +. Let =,,, Á,, find function values, and connect the resulting points with a smooth curve. Then create a graph for the function that models the percentage of adult height attained b a girl who is ears old, g = log - 4. Let = 5, 6, 7, Á, 5, find function values, and connect the resulting points with a smooth curve. Group members should then discuss similarities and differences in the growth patterns for bos and girls based on the graphs. SECTION. Properties of Logarithms Objectives ❶ ❷ ❸ ❹ ❺ ❻ Use the product rule. Use the quotient rule. Use the power rule. Epand logarithmic epressions. Condense logarithmic epressions. Use the change-of-base propert. We all learn new things in different was. In this section, we consider important properties of logarithms. What would be the most effective wa for ou to learn about these properties? Would it be helpful to use our graphing utilit and discover one of these properties for ourself? To do so, work Eercise in Eercise Set. before continuing. Would the properties become more meaningful if ou could see eactl where the come from? If so, ou will find details of the proofs of some of these properties in the appendi. The remainder of our work in this chapter will be based on the properties of logarithms that ou learn in this section. ❶ Use the product rule. The Product Rule Properties of eponents correspond to properties of logarithms. For eample, when we multipl with the same base, we add eponents: bm # bn = bm + n. This propert of eponents, coupled with an awareness that a logarithm is an eponent, suggests the following propert, called the product rule:

27 Section. Properties of Logarithms 40 Discover We know that log 00,000 = 5. Show that ou get the same result b writing 00,000 as 000 # 00 and then using the product rule. Then verif the product rule b using other numbers whose logarithms are eas to find. The Product Rule Let b, M, and N be positive real numbers with b Z. log b MN = log b M + log b N The logarithm of a product is the sum of the logarithms. When we use the product rule to write a single logarithm as the sum of two logarithms, we sa that we are epanding a logarithmic epression. For eample, we can use the product rule to epand ln7: ln (7) = ln 7 + ln. The logarithm of a product is the sum of the logarithms. EXAMPLE Using the Product Rule Use the product rule to epand each logarithmic epression: a. log 4 7 # 5 b. log0. Solution a. log 4 7 # 5 = log4 7 + log 4 5 The logarithm of a product is the sum of the logarithms. b. log0 = log 0 + log = + log The logarithm of a product is the sum of the logarithms. These are common logarithms with base 0 understood. Because log b b, then log 0. ❷Use the quotient rule. Discover We know that log 6 = 4. Show that ou get the same result b writing 6 as and then using the quotient rule. Then verif the quotient rule using other numbers whose logarithms are eas to find. Point Use the product rule to epand each logarithmic epression: a. log 6 7 # b. log00. The Quotient Rule When we divide with the same base, we subtract eponents: This propert suggests the following propert of logarithms, called the quotient rule: The Quotient Rule b m b n = bm - n. Let b, M, and N be positive real numbers with b Z. log b a M N b = log b M - log b N The logarithm of a quotient is the difference of the logarithms. When we use the quotient rule to write a single logarithm as the difference of two logarithms, we sa that we are epanding a logarithmic epression. For eample, we can use the quotient rule to epand log a b : log a b = log - log. The logarithm of a quotient is the difference of the logarithms.

28 40 Chapter Eponential and Logarithmic Functions EXAMPLE Using the Quotient Rule Use the quotient rule to epand each logarithmic epression: a. log 7a 9 b. b ln e 7. Solution a. b. log 7 a 9 b = log log 7 ln e 7 = ln e - ln 7 = - ln 7 The logarithm of a quotient is the difference of the logarithms. The logarithm of a quotient is the difference of the logarithms. These are natural logarithms with base e understood. Because ln e, then ln e. ❸Use the power rule. Point Use the quotient rule to epand each logarithmic epression: a. log 8a b. b The Power Rule ln e5. When an eponential epression is raised to a power, we multipl eponents: b m n = b mn. This propert suggests the following propert of logarithms, called the power rule: The Power Rule Let b and M be positive real numbers with b Z, and let p be an real number. log b M p = p log b M The logarithm of a number with an eponent is the product of the eponent and the logarithm of that number. When we use the power rule to pull the eponent to the front, we sa that we are epanding a logarithmic epression. For eample, we can use the power rule to epand ln : ln = ln. The logarithm of a number with an eponent is the product of the eponent and the logarithm of that number. Figure.6 shows the graphs of = ln and = ln in -5, 5, 4 b -5, 5, 4 viewing rectangles. Are ln and ln the same? The graphs illustrate that = ln and = ln have different domains. The graphs are onl the same if 7 0. Thus, we should write ln = ln for 7 0. = ln = ln Figure.6 ln and ln have different domains. Domain: (, 0) (0, ) Domain: (0, )

29 Section. Properties of Logarithms 40 When epanding a logarithmic epression, ou might want to determine whether the rewriting has changed the domain of the epression. For the rest of this section, assume that all variables and variable epressions represent positive numbers. EXAMPLE Using the Power Rule Use the power rule to epand each logarithmic epression: a. log b. c. log ln. Solution a. log = 4 log 5 7 The logarithm of a number with an eponent is the eponent times the logarithm of the number. b. ln = ln = ln Rewrite the radical using a rational eponent. Use the power rule to bring the eponent to the front. c. log4 5 = 5 log4 We immediatel appl the power rule because the entire variable epression, 4, is raised to the 5th power. ❹ Epand logarithmic epressions. Point Use the power rule to epand each logarithmic epression: a. log b. ln c. log Epanding Logarithmic Epressions It is sometimes necessar to use more than one propert of logarithms when ou epand a logarithmic epression. Properties for epanding logarithmic epressions are as follows: Properties for Epanding Logarithmic Epressions For M 7 0 and N 7 0:. log b MN = log b M + log b N Product rule. log b a M Quotient rule N b = log b M - log b N. log b M p = p log b M Power rule Stud Tip The graphs show that In general, log b M + N Z log b M + log b N. = ln ( + ) ln (+) ln +ln. = ln shifted units left = ln + ln = ln shifted ln units up Tr to avoid the following errors: Incorrect! log b M + N = log b M + log b N log b M - N = log b M - log b N log b M # N = logb M # logb N log b a M N b = log b M log b N log b M log b N = log b M - log b N log b MN p = p log b MN [ 4, 5, ] b [,, ]

30 404 Chapter Eponential and Logarithmic Functions EXAMPLE 4 Epanding Logarithmic Epressions Use logarithmic properties to epand each epression as much as possible: a. log b b. log Solution We will have to use two or more of the properties for epanding logarithms in each part of this eample. a. log b = log b Q R Use eponential notation. = log b + log b Use the product rule. = log b + log b Use the power rule. b. log = log Use eponential notation. = log 6 - log = log 6 - log log 6 4 = log 6 - log log 6 Use the quotient rule. Use the product rule on log Use the power rule. = log 6 - log log 6 = log log 6 Appl the distributive propert. log 6 6 because is the power to which we must raise 6 to get ❺ Condense logarithmic epressions. Stud Tip These properties are the same as those in the bo on page 40. The onl difference is that we ve reversed the sides in each propert from the previous bo. Point4 Use logarithmic properties to epand each epression as much as possible: a. log ba 4 B b. log 5 5. Condensing Logarithmic Epressions To condense a logarithmic epression, we write the sum or difference of two or more logarithmic epressions as a single logarithmic epression. We use the properties of logarithms to do so. Properties for Condensing Logarithmic Epressions For M 7 0 and N 7 0:. log b M + log b N = log b MN Product rule. log b M - log b N = log b a M Quotient rule N b. p log b M = log b M p Power rule EXAMPLE 5 Condensing Logarithmic Epressions Write as a single logarithm: a. log 4 + log 4 b. log4 - - log. Solution a. log 4 + log 4 = log 4 # Use the product rule. = log 4 64 We now have a single logarithm. However, we can simplif. = log 4 64 because 4 64.

31 Section. Properties of Logarithms 405 b. log4 - - log = loga 4 - b Use the quotient rule. Point5 Write as a single logarithm: a. log 5 + log 4 b. log log. Coefficients of logarithms must be before ou can condense them using the product and quotient rules. For eample, to condense ln + ln +, the coefficient of the first term must be. We use the power rule to rewrite the coefficient as an eponent:. Use the power rule to make the number in front an eponent. ln +ln (+)=ln +ln (+)=ln [ (+)].. Use the product rule. The sum of logarithms with coefficients of is the logarithm of the product. EXAMPLE 6 Condensing Logarithmic Epressions Write as a single logarithm: a. log + 4 log - b. ln ln c. 4 log b - log b 6 - log b. Solution a. log + 4 log - b. = log + log - 4 = logc - 4 D ln ln = ln ln = lnb + 7 R Use the power rule so that all coefficients are. Use the product rule. The condensed form can be epressed as log Use the power rule so that all coefficients are. Use the quotient rule. c. 4 log b - log b 6 - log b = log b 4 - log b 6 - log b = log b 4 - Alog b 6 + log b B = log b 4 - log b A6 B = log b 4 6 or log b 4 6 Use the power rule so that all coefficients are. Rewrite as a single subtraction. Use the product rule. Use the quotient rule. Point6 Write as a single logarithm: a. ln + ln + 5 b. log - - log c. 4 log b - log b 5-0 log b.

32 406 Chapter Eponential and Logarithmic Functions ❻ Use the change-of-base propert. The Change-of-Base Propert We have seen that calculators give the values of both common logarithms (base 0) and natural logarithms (base e). To find a logarithm with an other base, we can use the following change-of-base propert: The Change-of-Base Propert For an logarithmic bases a and b, and an positive number M, log b M = log a M log a b. The logarithm of M with base b is equal to the logarithm of M with an new base divided b the logarithm of b with that new base. In the change-of-base propert, base b is the base of the original logarithm. Base a is a new base that we introduce. Thus, the change-of-base propert allows us to change from base b to an new base a, as long as the newl introduced base is a positive number not equal to. The change-of-base propert is used to write a logarithm in terms of quantities that can be evaluated with a calculator. Because calculators contain kes for common (base 0) and natural (base e) logarithms, we will frequentl introduce base 0 or base e. Change-of-Base Introducing Common Introducing Natural Propert Logarithms Logarithms log b M= log a M log a b log b M= log 0 M log 0 b log b M= log e M log e b a is the new introduced base. 0 is the new introduced base. e is the new introduced base. Using the notations for common logarithms and natural logarithms, we have the following results: The Change-of-Base Propert: Introducing Common and Natural Logarithms Introducing Common Logarithms Introducing Natural Logarithms log b M = log M log b log b M = ln M ln b Discover Find a reasonable estimate of log 5 40 to the nearest whole number. To what power can ou raise 5 in order to get 40? Compare our estimate to the value obtained in Eample 7. EXAMPLE 7 Changing Base to Common Logarithms Use common logarithms to evaluate log Solution Because log b M = log M log b, log 5 40 = This means that log 5 40 L.07. log 40 log 5 L.07. LOG LOG Use a calculator: 40 5 or LOG 40 LOG 5 ENTER. On some calculators, parentheses are needed after 40 and 5. Point7 Use common logarithms to evaluate log

33 EXAMPLE 8 Section. Properties of Logarithms 407 Changing Base to Natural Logarithms Use natural logarithms to evaluate log Solution Because log b M = ln M ln b, log 5 40 = ln 40 ln 5 L.07. We have again shown that log 5 40 L.07. LN LN Use a calculator: 40 5 or LN 40 LN 5 ENTER. On some calculators, parentheses are needed after 40 and 5. Point8 Use natural logarithms to evaluate log Technolog We can use the change-of-base propert to graph logarithmic functions with bases other than 0 or e on a graphing utilit. For eample, Figure.7 shows the graphs of = log and = log 0 in a 0, 0, 4 b -,, 4 viewing rectangle. Because log = ln and ln the functions are entered as log 0 = ln ln 0, = log 0 = log and = LN LN = LN LN 0. On some calculators, parentheses are needed after,, and 0. Figure.7 Using the change-ofbase propert to graph logarithmic functions EXERCISE SET. Practice Eercises In Eercises 40, use properties of logarithms to epand each logarithmic epression as much as possible. Where possible, evaluate logarithmic epressions without using a calculator.. log5 7 #. log 8 # 7. log log log log 0, log 7a 7 8. log 9a 9 9. b b 0. loga. log 4a b b. e ln 5 4. ln e log b 6. log b 7 7. log N log M -8 loga 00 b log 5 a 5 b 9. ln5 0. ln7. log b. log b. log log 6 6. log log b 9. log00 0. lne z... log b log log A A log b 5. log 5 6. log 5 A 5 A 6 z 5 7. lnb + R lnb 4 + R + 5 log 5 5 log b z logb logb 0 - R R z

34 408 Chapter Eponential and Logarithmic Functions z In Eercises 4 70, use properties of logarithms to condense each logarithmic epression. Write the epression as a single logarithm whose coefficient is. Where possible, evaluate logarithmic epressions without using a calculator. 4. log 5 + log 4. log 50 + log 4 4. ln + ln ln + ln 45. log 96 - log 46. log log log log 48. log log 49. log + log 50. log + 7 log 5. ln + ln 5. ln + ln 5. log b + log b log b + 6 log b ln - ln ln - ln 57. ln - ln 58. ln - ln ln ln ln ln 6. ln + 5 ln - 6 ln z 6. 4 ln + 7 ln - ln 6. log + log 64. log 4 - log log 5 + log 5 - log 5 + log 4 - log 4 + log ln ln - ln ln ln - ln log + log - - log 7 - log + log + log log 5 - log + In Eercises 7 78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. 7. log 5 7. log log log log log log p log p 400 In Eercises 79 8, use a graphing utilit and the change-of-base propert to graph each function. 79. = log 80. = log 5 8. = log + 8. = log - Practice Plus In Eercises 8 88, let log b = A and log b = C. Write each epression in terms of A and C. 8. log b 84. log b log b log b log b 88. log A 7 b A 6 In Eercises 89 0, determine whether each equation is true or false. Where possible, show work to support our conclusion. If the statement is false, make the necessar change(s) to produce a true statement. 89. ln e = ln 0 = e 9. log 4 = log 4 9. ln8 = ln 9. log 0 = 94. ln + = ln + ln 95. ln5 + ln = ln5 96. ln + ln = ln log + log + - log = log log + = log + - log - log log 6 a b = log log log = log log log 0. e = 7 = log 7 ln Application Eercises 0. The loudness level of a sound can be epressed b comparing the sound s intensit to the intensit of a sound barel audible to the human ear. The formula D = 0log I - log I 0 describes the loudness level of a sound, D, in decibels, where I is the intensit of the sound, in watts per meter, and I 0 is the intensit of a sound barel audible to the human ear. a. Epress the formula so that the epression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question: If a sound has an intensit 00 times the intensit of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound? 04. The formula t = ln A - lna - N4 c describes the time, t, in weeks, that it takes to achieve master of a portion of a task, where A is the maimum learning possible, N is the portion of the learning that is to be achieved, and c is a constant used to measure an individual s learning stle. a. Epress the formula so that the epression in brackets is written as a single logarithm. b. The formula is also used to determine how long it will take chimpanzees and apes to master a task. For eample, a tpical chimpanzee learning sign language can master a maimum of 65 signs. Use the form of the formula from part (a) to answer this question: How man weeks will it take a chimpanzee to master 0 signs if c for that chimp is 0.0? Writing in Mathematics 05. Describe the product rule for logarithms and give an eample. 06. Describe the quotient rule for logarithms and give an eample. 07. Describe the power rule for logarithms and give an eample. 08. Without showing the details, eplain how to condense ln - ln Describe the change-of-base propert and give an eample. 0. Eplain how to use our calculator to find log You overhear a student talking about a propert of logarithms in which division becomes subtraction. Eplain what the student means b this.. Find ln using a calculator. Then calculate each of the following: ; ; 4 ; ; Á. Describe what ou observe.

35 Mid-Chapter Point 409 Technolog Eercises. a. Use a graphing utilit (and the change-of-base propert) to graph = log. b. Graph = + log, = log +, and = -log in the same viewing rectangle as = log. Then describe the change or changes that need to be made to the graph of = log to obtain each of these three graphs. 4. Graph = log, = log0, and = log0. in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic propert accounts for this relationship? 5. Use a graphing utilit and the change-of-base propert to graph = log, = log 5, and = log 00 in the same viewing rectangle. a. Which graph is on the top in the interval 0,? Which is on the bottom? b. Which graph is on the top in the interval, q? Which is on the bottom? c. Generalize b writing a statement about which graph is on top, which is on the bottom, and in which intervals, using = log b where b 7. Disprove each statement in Eercises 6 0 b first letting equal a positive constant of our choice, and then using a graphing utilit to graph the function on each side of the equal sign. The two functions should have different graphs, showing that the equation is not true in general. 6. log + = log + log CHAPTER MID-CHAPTER CHECK POINT What You Know: We evaluated and graphed eponential functions [ f = b, b 7 0 and b Z ], including the natural eponential function f = e, e L.784. A function has an inverse that is a function if there is no horizontal line that intersects the function s graph more than once. The eponential function passes this horizontal line test and we called the inverse of the eponential function with base b the logarithmic function with base b. We learned that = log is equivalent to b b =. We evaluated and graphed logarithmic functions, including the common logarithmic function f = log 0 or f = log 4 and the natural logarithmic function f = log e or f = ln 4. We learned to use transformations to graph eponential and logarithmic functions. Finall, we used properties of logarithms to epand and condense logarithmic epressions. In Eercises 5, graph f and g in the same rectangular coordinate sstem. Graph and give equations of all asmptotes. Give each function s domain and range.. f = and g = -. f = A B and g = A B -. f = e and g = ln 4. f = log and g = log g = log and g = - log 7. log 8. ln - = ln - ln = log log 9. ln = ln ln 0. Critical Thinking Eercises. Which one of the following is true? a. log 7 49 log 7 7 = log log 7 7 b. log b + = log b + log b c. log b 5 = log b + log b 5 d. ln = ln. Use the change-of-base propert to prove that. If log = A and log 7 = B, find log 7 9 in terms of A and B. 4. Write as a single term that does not contain a logarithm: 5. If f = log b, show that f + h - f h log e = ln ln ln 0. e ln 85 - ln. = ln - ln = log b a + h b h, h Z 0. In Eercises 6 9, find the domain of each function. 6. f = log g = log h = log f = + 6 In Eercises 0 0, evaluate each epression without using a calculator. If evaluation is not possible, state the reason. 0. log 8 + log 5 5. log 9. log log 0 4. log log 8 5. log Alog 8 B 6. 6 log ln e 8. 0 log 9. log log p p p In Eercises, epand and evaluate numerical terms.. log 000. lne 9 0 In Eercises 5, write each epression as a single logarithm.. 8 log log 5 + log 5 log 7 5. ln - ln - lnz - 6. Use the formulas A = Pa + r nt n b and A = Pe rt to solve this eercise. You decide to invest $8000 for ears at an annual rate of 8%. How much more is the return if the interest is compounded continuousl than monthl? Round to the nearest dollar.

36 pr i-hr 40 /6/06 :04 PM Page 40 Chapter Eponential and Logarithmic Functions SECTION.4 Eponential and Logarithmic Equations Objectives ❶ ❷ ❸ ❹ ❺ Use like bases to solve eponential equations. Use logarithms to solve eponential equations. Use the definition of a logarithm to solve logarithmic equations. Use the one-to-one propert of logarithms to solve logarithmic equations. Solve applied problems involving eponential and logarithmic equations. ❶ Use like bases to solve eponential equations. You inherited $0,000. You would like to put aside $5,000 and eventuall have over half a million dollars for earl retirement. Is this possible? In this section, ou will see how techniques for solving equations with variable eponents provide an answer to the question. Eponential Equations An eponential equation is an equation containing a variable in an eponent. Eamples of eponential equations include - 8 = 6, 4 = 5, and 40e0.6 = 40. Some eponential equations can be solved b epressing each side of the equation as a power of the same base. All eponential functions are one-to-one that is, no two different ordered pairs have the same second component. Thus, if b is a positive number other than and bm = bn, then M = N. Solving Eponential Equations b Epressing Each Side as a Power of the Same Base If bm = bn, then M=N. Epress each side as a power of the same base.. Rewrite the equation in the form bm = bn.. Set M = N.. Solve for the variable. Technolog The graphs of = - 8 and = 6 EXAMPLE have an intersection point whose -coordinate is 4. This verifies that 546 is the solution set of - 8 = 6. Solve: a. - 8 Solving Eponential Equations = 6 b. 7 + = 9 -. Solution In each equation, epress both sides as a power of the same base. Then set the eponents equal to each other and solve for the variable. a. Because 6 is 4, we epress each side of - 8 = 6 in terms of base. = 6 = 8 =4 [, 5, ] b [ 0, 0, ] Set the eponents equal to each other. - 8 = 6 This is the given equation = Write each side as a power of the same base. - 8 = 4 If bm bn, b>0 and b, then M N. = Add 8 to both sides. = 4 Divide both sides b. Substituting 4 for into the original equation produces the true statement 6 = 6. The solution set is 546.

37 Section.4 Eponential and Logarithmic Equations 4 b. Because 7 = and 9 =, we can epress both sides of 7 + = 9 - in terms of base. 7 + = = = + = = = - = - This is the given equation. Write each side as a power of the same base. When an eponential epression is raised to a power, multipl eponents. If two powers of the same base are equal, then the eponents are equal. Appl the distributive propert. Subtract from both sides. Subtract 9 from both sides. Substituting - for into the original equation produces 7-8 = 9 -, which simplifies to the true statement -4 = -4. The solution set is 5-6. Point Solve: a. 5-6 = 5 b. 8 + = 4 -. ❷ Use logarithms to solve eponential equations. Most eponential equations cannot be rewritten so that each side has the same base. Logarithms are etremel useful in solving such equations. The solution begins with isolating the eponential epression and taking the natural logarithm on both sides. Wh can we do this? All logarithmic relations are functions. Thus, if M and N are positive real numbers and M = N, then log b M = log b N. Using Natural Logarithms to Solve Eponential Equations. Isolate the eponential epression.. Take the natural logarithm on both sides of the equation.. Simplif using one of the following properties: 4. Solve for the variable. ln b = ln b or ln e =. Discover The base that is used when taking the logarithm on both sides of an equation can be an base at all. Solve 4 = 5 b taking the common logarithm on both sides. Solve again, this time taking the logarithm with base 4 on both sides. Use the change-ofbase propert to show that the solutions are the same as the one obtained in Eample. EXAMPLE Solve: 4 = 5. Solving an Eponential Equation Solution Because the eponential epression, 4, is alread isolated on the left, we begin b taking the natural logarithm on both sides of the equation. 4 = 5 ln 4 = ln 5 ln 4 = ln 5 = ln 5 ln 4 This is the given equation. Take the natural logarithm on both sides. Use the power rule and bring the variable eponent to the front: ln b ln b. Solve for b dividing both sides b ln 4. We now have an eact value for. We use the eact value for in the equation s ln 5 ln 5 solution set. Thus, the equation s solution is and the solution set is e ln 4 ln 4 f. We can obtain a decimal approimation b using a calculator: L.95. Because 4 = 6, it seems reasonable that the solution to 4 = 5 is approimatel.95.

38 4 Chapter Eponential and Logarithmic Functions Point Solve: 5 = 4. Find the solution set and then use a calculator to obtain a decimal approimation to two decimal places for the solution. EXAMPLE Solve: 40e = 7. Solving an Eponential Equation Solution We begin b adding to both sides and dividing both sides b 40 to isolate the eponential epression, e 0.6. Then we take the natural logarithm on both sides of the equation. 40e = 7 40e 0.6 = 40 e 0.6 = 6 ln e 0.6 = ln = ln 6 = ln L.99 Thus, the solution of the equation is solution in the original equation to verif that This is the given equation. Add to both sides. Isolate the eponential factor b dividing both sides b 40. Take the natural logarithm on both sides. Use the inverse propert ln e on the left. Divide both sides b 0.6 and solve for. ln L.99. e ln f Tr checking this approimate is the solution set. Point Solve: 7e - 5 = 58. Find the solution set and then use a calculator to obtain a decimal approimation to two decimal places for the solution. EXAMPLE 4 Solving an Eponential Equation Solve: 5 - = 4 +. Solution Because each eponential epression is isolated on one side of the equation, we begin b taking the natural logarithm on both sides. 5 - = 4 + This is the given equation. ln 5 - = ln 4 + Take the natural logarithm on both sides. Be sure to insert parentheses around the binomials. (-) ln 5=(+) ln 4 Remember that ln 5 and ln 4 are constants, not variables. Use the power rule and bring the variable eponents to the front: ln b ln b. Discover Use properties of logarithms to show that the solution in Eample 4 can be epressed as ln 600. lna 5 6 B ln 5 - ln 5 = ln 4 + ln 4 ln 5 - ln 4 = ln 5 + ln 4 ln 5 - ln 4 = ln 5 + ln 4 = ln 5 + ln 4 ln 5 - ln 4 Use the distributive propert to distribute ln 5 and ln 4 to both terms in parentheses. Collect variable terms involving on the left b subtracting ln 4 and adding ln 5 on both sides. Factor out from the two terms on the left. Isolate b dividing both sides b ln 5 ln 4. ln 5 + ln 4 The solution set is e The solution is approimatel ln 5 - ln 4 f. Point4 Solve: - = 7 +. Find the solution set and then use a calculator to obtain a decimal approimation to two decimal places for the solution.

39 Section.4 Eponential and Logarithmic Equations 4 Technolog Shown below is the graph of = e - 4e +. There are two -intercepts, one at 0 and one at approimatel.0. These intercepts verif our algebraic solution. EXAMPLE 5 Solve: e - 4e + = 0. Solving an Eponential Equation Solution The given equation is quadratic in form. If u = e, the equation can be epressed as u - 4u + = 0. Because this equation can be solved b factoring, we factor to isolate the eponential term. e - 4e + = 0 e - e - = 0 e - = 0 or e - = 0 e = e = ln e = ln = 0 = ln This is the given equation. Factor on the left. Notice that if u e, u 4u u u. Set each factor equal to 0. Solve for e. Take the natural logarithm on both sides of the first equation. The equation on the right can be solved b inspection. ln e The solution set is 50, ln 6. The solutions are 0 and ln, which is approimatel.0. [,, ] b [,, ] ❸ Use the definition of a logarithm to solve logarithmic equations. Point5 Solve: e - 8e + 7 = 0. Find the solution set and then use a calculator to obtain a decimal approimation to two decimal places, if necessar, for the solutions. Logarithmic Equations A logarithmic equation is an equation containing a variable in a logarithmic epression. Eamples of logarithmic equations include log 4 + = and ln + - ln4 + = lna b. Some logarithmic equations can be epressed in the form log b M = c. We can solve such equations b rewriting them in eponential form. Using the Definition of a Logarithm to Solve Logarithmic Equations. Epress the equation in the form log b M = c.. Use the definition of a logarithm to rewrite the equation in eponential form: log b M=c means b c =M. Logarithms are eponents.. Solve for the variable. 4. proposed solutions in the original equation. Include in the solution set onl values for which M 7 0. EXAMPLE 6 Solving Logarithmic Equations Solve: a. log 4 + = b. ln =. Solution The form log b M = c involves a single logarithm whose coefficient is on one side and a constant on the other side. Equation (a) is alread in this form.we will need to divide both sides of equation (b) b to obtain this form.

40 44 Chapter Eponential and Logarithmic Functions Technolog The graphs of = log 4 + and = have an intersection point whose -coordinate is. This verifies that 56 is the solution set for log 4 + =. = = = log 4 ( + ) [, 7, ] b [,, ] a. log 4 + = This is the given equation. : Rewrite in eponential form: log means b c b M c M. Square 4. Subtract from both sides. This is the given logarithmic equation. Substitute for. log because This true statement indicates that the solution set is 56. b. ln = This is the given equation. ln = 4 log e = 4 4 = + 6 = + = log 4 + = log 4 + log 4 6 e 4 = =, true Divide both sides b. Rewrite the natural logarithm showing base e. This step is optional. Rewrite in eponential form: log means b c b M c M. e 4 = e 4 : ln = lnb e4 R ln e 4 # 4 =, true Divide both sides b. This is the given logarithmic equation. e 4 Substitute for. # Simplif: e4 e 4. Because ln e, we conclude ln e 4 4. This true statement indicates that the solution set is b e4 r. Point6 Solve: a. log - 4 = b. 4 ln = 8. Logarithmic epressions are defined onl for logarithms of positive real numbers. Alwas check proposed solutions of a logarithmic equation in the original equation. Eclude from the solution set an proposed solution that produces the logarithm of a negative number or the logarithm of 0. To rewrite the logarithmic equation log b M = c in the equivalent eponential form b c = M, we need a single logarithm whose coefficient is one. It is sometimes necessar to use properties of logarithms to condense logarithms into a single logarithm. In the net eample, we use the product rule for logarithms to obtain a single logarithmic epression on the left side.

41 Section.4 Eponential and Logarithmic Equations 45 EXAMPLE 7 Solving a Logarithmic Equation Solve: log + log - 7 =. Solution log + log - 7 = log - 74 = = = = = = 0 or + = 0 = 8 = - 8: log + log - 7 = log 8 + log 8-7 log 8 + log + 0 =, true The solution set is 586. This is the given equation. Use the product rule to obtain a single logarithm: log b M log b N log b MN. Rewrite in eponential form: log means b c b M c M. Evaluate on the left and appl the distributive propert on the right. Set the equation equal to 0. Factor. Set each factor equal to 0. Solve for. : log + log - 7 = log - + log The number does not check. Negative numbers do not have logarithms. ❹ Use the one-to-one propert of logarithms to solve logarithmic equations. Point7 Solve: log + log - =. Some logarithmic equations can be epressed in the form log b M = log b N. Because all logarithmic functions are one-to-one, we can conclude that M = N. Using the One-to-One Propert of Logarithms to Solve Logarithmic Equations. Epress the equation in the form log b M = log b N. This form involves a single logarithm whose coefficient is on each side of the equation.. Use the one-to-one propert to rewrite the equation without logarithms: If log b M = log b N, then M = N.. Solve for the variable. 4. proposed solutions in the original equation. Include in the solution set onl values for which M 7 0 and N 7 0. EXAMPLE 8 Solving a Logarithmic Equation Solve: ln + - ln4 + = lna b. Solution In order to appl the one-to-one propert of logarithms, we need a single logarithm whose coefficient is on each side of the equation. The right side is alread in this form. We can obtain a single logarithm on the left side b appling the quotient rule.

42 46 Chapter Eponential and Logarithmic Functions ln + - ln4 + = lna b This is the given equation. Technolog TABLE A graphing utilit s feature can be used to verif that 56 is the solution set of ln + - ln4 + = lna b. = ln( + ) ln(4 + ) = ln( ) and are equal when =. lna b = lna b = 4 + a b = 4 + a b + = = = = 0 - = 0 or + = 0 = = - Use the quotient rule to obtain a single logarithm on the left side: log b M log b N log b a M N b. Use the one-to-one propert: If log b M log b N, then M N. Multipl both sides b 4, the LCD. Simplif. Appl the distributive propert. Subtract 4 from both sides and set the equation equal to 0. Factor. Set each factor equal to 0. Solve for. Substituting for into the original equation produces the true statement lna B = lna B. However, substituting - produces logarithms of negative numbers. Thus, - is not a solution. The solution set is 56. ❺ Solve Visualizing the Relationship between Blood Alcohol Concentration and the Risk of a Car Accident A blood alcohol concentration of 0. corresponds to near certaint, or a 00% probabilit, of a car accident. Risk of a Car Accident applied problems involving eponential and logarithmic equations. 00% 80% 60% 40% R 0% R = 6e Blood Alcohol Concentration Point8 Solve: Applications Our first applied eample provides a mathematical perspective on the old slogan Alcohol and driving don t mi. In California, where 8% of fatal traffic crashes involve drinking drivers, it is illegal to drive with a blood alcohol concentration of 0.08 or higher.at these levels, drivers ma be arrested and charged with driving under the influence. EXAMPLE 9 Alcohol and Risk of a Car Accident Medical research indicates that the risk of having a car accident increases eponentiall as the concentration of alcohol in the blood increases.the risk is modeled b where is the blood alcohol concentration and R, given as a percent, is the risk of having a car accident. What blood alcohol concentration corresponds to a 0% risk of a car accident? Solution For a risk of 0% we let R = 0 in the equation and solve for, the blood alcohol concentration. R = 6e.77 6e.77 = 0 e.77 = 0 6 ln e.77 = lna 0 6 b.77 = lna 0 6 b = ln - = ln7 - - ln +. lna 0 6 b.77 L 0.09 R = 6e.77, This is the given equation. Substitute 0 for R and (optional) reverse the two sides of the equation. Isolate the eponential factor b dividing both sides b 6. Take the natural logarithm on both sides. Use the inverse propert ln e on the left. Divide both sides b.77 and solve for.

43 Section.4 Eponential and Logarithmic Equations 47 For a blood alcohol concentration of 0.09, the risk of a car accident is 0%. In man states, it is illegal to drive at 0.08, which is below this blood alcohol concentration. Point9 Use the formula in Eample 9 to answer this question:what blood alcohol concentration corresponds to a 7% risk of a car accident? (In man states, drivers under the age of can lose their licenses for driving at this level.) Suppose that ou inherit $0,000. Is it possible to invest $5,000 and have over half a million dollars for earl retirement? Our net eample illustrates the power of compound interest. EXAMPLE 0 The formula Revisiting the Formula for Compound Interest A = Pa + r n b nt describes the accumulated value, A, of a sum of mone, P, the principal, after t ears at annual percentage rate r (in decimal form) compounded n times a ear. How long will it take $5,000 to grow to $500,000 at 9% annual interest compounded monthl? Solution A = Pa + r n b nt 500,000 = 5,000a b t This is the given formula. Athe desired accumulated value $500,000, Pthe principal $5,000, rthe interest rate 9% 0.09, and n (monthl compounding). Plaing Doubles: Interest Rates and Doubling Time One wa to calculate what our savings will be worth at some point in the future is to consider doubling time. The following table shows how long it takes for our mone to double at different annual interest rates subject to continuous compounding. Annual Interest Years to Rate Double 5%.9 ears 7% 9.9 ears 9% 7.7 ears % 6. ears Of course, the first problem is collecting some mone to invest. The second problem is finding a reasonabl safe investment with a return of 9% or more. Our goal is to solve the equation for t. Let s reverse the two sides of the equation and then simplif within parentheses. 5,000a b t = 500,000 5, t = 500,000 5, t = 500,000 Reverse the two sides of the previous equation. Divide within parentheses: Add within parentheses. Divide both sides b 5,000. Take the natural logarithm on both sides. Use the power rule to bring the eponent to the front: ln M p p ln M. Solve for t, dividing both sides b ln Use a calculator. After approimatel.4 ears, the $5,000 will grow to an accumulated value of $500,000. If ou set aside the mone at age 0, ou can begin enjoing a life of leisure at about age 5. Point t = 0 ln.0075 t = ln 0 t ln.0075 = ln 0 t = ln 0 ln.0075 L How long, to the nearest tenth of a ear, will it take $000 to grow to $600 at 8% annual interest compounded quarterl?

44 48 Chapter Eponential and Logarithmic Functions EXAMPLE The Growth in the Number of U.S. Internet Users The bar graph in Figure.8 shows the number, in millions, of Internet users in the United States from 000 through 00. The function f = 4. ln models the number of U.S. Internet users, f, in millions, ears after 999. B which ear will there be 00 million Internet users in the United States? Solution We substitute 00 for f and solve for, the number of ears after 999. f = 4. ln = 4. ln This is the given function. Substitute 00 for f. Our goal is to isolate ln in the equation 00 = 4. ln We can then find b using the definition of a logarithm to rewrite the equation in eponential form. 4. ln = ln = 8. ln = log e = e 4. = L Number of Internet Users (millions) Reverse the two sides of the equation. Subtract 7.7 from both sides. Divide both sides b 4.. Rewrite the natural logarithm showing base e. This step is optional. Rewrite in eponential form: log b M c means b c M. Use a calculator. Approimatel ears after 999, in the ear 00, there will be 00 million Internet users in the United States Figure Source: Jupiter Media Number of Internet Users in the U.S Year Point Use the function in Eample to find in which ear there will be 0 million Internet users in the United States. EXERCISE SET.4 Practice Eercises Solve each eponential equation in Eercises b epressing each side as a power of the same base and then equating eponents.. = 64. = 8. 5 = = = 6. + = = = 5 9. = =. 9 = 7. 5 = = = = = 8. 9 = = = 4 +. e + =. e = 7 e + 4 = e Solve each eponential equation in Eercises 48. Epress the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approimation, correct to two decimal places, for the solution.. 0 = = 8.07

45 Section.4 Eponential and Logarithmic Equations e = e = = = e = 0. 9e = e 5 = e 7 = 0,7 8.. e - 5 = e - 8 = e = 0, e =, = = = = = = + 4. e - e + = e - e - = e 4 + 5e - 4 = e 4 - e - 8 = = = log = log 5 log log = log5 + log log = log7 + log - log 7 = log log - + log 5 = log 00 log + log + = log 0 log + + log - = log 4 ln ln + = ln - 8 log - - log + = log a b ln - - ln + = ln - - ln + 7 ln ln + 4 = ln - - ln + Solve each logarithmic equation in Eercises Be sure to reject an value of that is not in the domain of the original logarithmic epressions. Give the eact answer. Then, where necessar, use a calculator to obtain a decimal approimation, correct to two decimal places, for the solution. 49. log = log 5 = 5. ln = 5. ln = 5. log = 54. log 5-7 = 55. log - 4 = log 7 + = log 4 + = 58. log 4 + = ln = ln = ln = ln = 6 6. ln + = 64. ln + 4 = log 5 + log = log log 6 = log log + = log - + log + = log + - log - 5 = log log 4 - = log + 4 = log 9 + log - = 5 - log 4 log log log = log - + log - log + = log + 4 = log + log 4 log5 + = log + + log log - = log + + log 4 log - = log + + log 79. log = log 5 Practice Plus In Eercises 9 00, solve each equation # 5 4 = # = 8 9. ƒln ƒ - 6 = ƒlog ƒ - 6 = = = ln + + ln - - ln = 0 ln - ln ln = = = 9 Application Eercises Use the formula R = 6e.77, where is the blood alcohol concentration and R, given as a percent, is the risk of having a car accident, to solve Eercises What blood alcohol concentration corresponds to a 5% risk of a car accident? 0. What blood alcohol concentration corresponds to a 50% risk of a car accident? 0. The formula A = 8.9e t models the population of New York State, A, in millions, t ears after 000. a. What was the population of New York in 000? b. When will the population of New York reach 9.6 million? 04. The formula A = 5.9e 0.05t models the population of Florida, A, in millions, t ears after 000. a. What was the population of Florida in 000? b. When will the population of Florida reach 9. million? In Eercises 05 08, complete the table for a savings account subject to n compoundings earl ca = Pa + r nt Round answers to n b d. one decimal place. Number of Time t Amount Compounding Annual Interest Accumulated in Invested Periods Rate Amount Years 05. $, % $0, $ % $5, $ $ $ $9000 4

46 40 Chapter Eponential and Logarithmic Functions In Eercises 09, complete the table for a savings account subject to continuous compounding A = Pe rt. Round answers to one decimal place. Amount Annual Interest Accumulated Time t Invested Rate Amount in Years 09. $8000 8% Double the amount invested 0. $8000 $,000. $50 Triple the amount invested 7. $7,45 4.5% $5,000. Fed up with junk mail clogging our computer? Despite high-profile legislation and lawsuits, the bar graph shows that spam has flourished. Percentage of Total U.S. Population Percentage of Inbound Considered Spam 90% 75% 60% 45% 0% 5% Spam Slam: Percentage of Inbound in the U.S. Considered Spam 5% 00 Source: Meta Group 5% 0% 5% 0% 5% 8% 980 4% 6% Year 6% 990 Year 6% % 004 The function f = ln models the percentage of inbound in the United States considered spam, f, ears after 000. a. How well does the function model the data for 00? b. If law enforcement against spammers does not change and the model is projected into the future, when will 96% of inbound be spam? Round to the nearest ear. 4. The bar graph shows the number of children under 8 as a percentage of the total U.S. population. 0% Number of Children Under 8 as a Percentage of Total U.S. Population 4% 970 Source: Projected 4% 00 The function f = ln models the number of children under 8 as a percentage of the total U.S. population, f, ears after 969. a. How well does the function model the projected data for 00? b. According to the model, when will children under 8 decline to % of the total U.S. population? Round to the nearest ear. The function P = 95-0 log models the percentage, P, of students who could recall the important features of a classroom lecture as a function of time, where represents the number of das that have elapsed since the lecture was given. The figure shows the graph of the function. Use this information to solve Eercises 5 6. Round answers to one decimal place. Percentage Remembering the Lecture P() = 95 0 log Das after Lecture 5. After how man das do onl half the students recall the important features of the classroom lecture? (Let P = 50 and solve for.) Locate the point on the graph that conves this information. 6. After how man das have all students forgotten the important features of the classroom lecture? (Let P = 0 and solve for.) Locate the point on the graph that conves this information. The ph of a solution ranges from 0 to 4. An acid solution has a ph less than 7. Pure water is neutral and has a ph of 7. Normal, unpolluted rain has a ph of about 5.6. The ph of a solution is given b ph = -log, where represents the concentration of the hdrogen ions in the solution, in moles per liter. Use the formula to solve Eercises An environmental concern involves the destructive effects of acid rain. The most acidic rainfall ever had a ph of.4. What was the hdrogen ion concentration? Epress the answer as a power of 0 and then round to the nearest thousandth.

47 Section.4 Eponential and Logarithmic Equations 4 8. The figure shows ver acidic rain in the northeast United States. What is the hdrogen ion concentration of rainfall with a ph of 4.? Epress the answer as a power of 0 and then round to the nearest hundred-thousandth. 5.4 Acid Rain over Canada and the United States Source: National Atmospheric Program Ver high acidit area: ph 4. or less High acidit area: ph 4.4 or less Hurricanes are one of nature s most destructive forces.these low-pressure areas often have diameters of over 500 miles. The function f = 0.48 ln models the barometric air pressure, f, in inches of mercur, at a distance of miles from the ee of a hurricane. Use this function to solve Eercises.. Graph the function in a 0, 500, 504 b 7, 0, 4 viewing rectangle. What does the shape of the graph indicate about barometric air pressure as the distance from the ee increases?. Use an equation to answer this question: How far from the ee of a hurricane is the barometric air pressure 9 inches of mercur? Use the TRACE and ZOOM features or the intersect command of our graphing utilit to verif our answer.. The function Pt = 45e -0.09t models a runner s pulse, Pt, in beats per minute, t minutes after a race, where 0 t 5. Graph the function using a graphing utilit. TRACE along the graph and determine after how man minutes the runner s pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verif our observation algebraicall. 4. The function Wt = e t models the weight, Wt, in kilograms, of a female African elephant at age t ears. kilogram L. pounds Use a graphing utilit to graph the function. Then TRACE along the curve to estimate the age of an adult female elephant weighing 800 kilograms. Writing in Mathematics 9. Eplain how to solve an eponential equation when both sides can be written as a power of the same base. 0. Eplain how to solve an eponential equation when both sides cannot be written as a power of the same base. Use = 40 in our eplanation.. Eplain the differences between solving log - = 4 and log - = log 4.. In man states, a 7% risk of a car accident with a blood alcohol concentration of 0.08 is the lowest level for charging a motorist with driving under the influence. Do ou agree with the 7% risk as a cutoff percentage, or do ou feel that the percentage should be lower or higher? Eplain our answer. What blood alcohol concentration corresponds to what ou believe is an appropriate percentage? Technolog Eercises In Eercises 0, use our graphing utilit to graph each side of the equation in the same viewing rectangle. Then use the -coordinate of the intersection point to find the equation s solution set. Verif this value b direct substitution into the equation.. + = = log 4-7 = log - = log + + log = log log = = = + 4 Critical Thinking Eercises 5. Which one of the following is true? a. If log + =, then e = +. b. If log7 + - log + 5 = 4, then the equation in eponential form is 0 4 = c. If = ln, then = e k. k d. Eamples of eponential equations include 0 = 5.7, e = 0.7, and 0 = If $4000 is deposited into an account paing % interest compounded annuall and at the same time $000 is deposited into an account paing 5% interest compounded annuall, after how long will the two accounts have the same balance? Round to the nearest ear. Solve each equation in Eercises 7 9. each proposed solution b direct substitution or with a graphing utilit. 7. ln = ln 8. log log + = 6 9. lnln = 0 Group Eercise 40. Research applications of logarithmic functions as mathematical models and plan a seminar based on our group s research. Each group member should research one of the following areas or an other area of interest: ph (acidit of solutions), intensit of sound (decibels), brightness of stars, human memor, progress over time in a sport, profit over time. For the area that ou select, eplain how logarithmic functions are used and provide eamples.

48 pr i-hr 4 /6/06 :04 PM Page 4 Chapter Eponential and Logarithmic Functions SECTION.5 Eponential Growth and Deca; Modeling Data Objectives ❶ ❷ ❸ ❹ ❺ Model eponential growth and deca. Use logistic growth models. Use Newton s Law of Cooling. Model data with eponential and logarithmic functions. Epress an eponential model in base e. The most casual cruise on the Internet shows how people disagree when it comes to making predictions about the effects of the world s growing population. Some argue that there is a recent slowdown in the growth rate, economies remain robust, and famines in North Korea and Ethiopia are aberrations rather than signs of the future. Others sa that the 6. billion people on Earth is twice as man as can be supported in middle-class comfort, and the world is running out of arable land and fresh water. Debates about entities that are growing eponentiall can be approached mathematicall: We can create functions that model data and use these functions to make predictions. In this section, we will show ou how this is done. ❶ Model eponential growth and deca. Eponential Growth and Deca One of algebra s man applications is to predict the behavior of variables. This can be done with eponential growth and deca models. With eponential growth or deca, quantities grow or deca at a rate directl proportional to their size. Populations that are growing eponentiall grow etremel rapidl as the get larger because there are more adults to have offspring. For eample, the growth rate for world population is.%, or 0.0. This means that each ear world population is.% more than what it was in the previous ear. In 00, world population was approimatel 6. billion. Thus, we compute the world population in 00 as follows: 6. billion + % of 6. billion = = This computation suggests that billion people populated the world in 00. The billion represents an increase of 80.6 million people from 00 to 00, the equivalent of the population of German. Using.% as the annual growth rate, world population for 00 is found in a similar manner: % of = L 6.6. This computation suggests that approimatel 6.6 billion people populated the world in 00. The eplosive growth of world population ma remind ou of the growth of mone in an account subject to compound interest. Just as the growth rate for world population is multiplied b the population plus an increase in the population, a compound interest rate is multiplied b our original investment plus an accumulated interest. The balance in an account subject to continuous compounding and world population are special cases of eponential growth models.

49 Section.5 Eponential Growth and Deca; Modeling Data 4 Stud Tip You have seen the formula for eponential growth before, but with different letters. It is the formula for compound interest with continous compounding. Amount at time t A = Pe rt Principal is the original amount. A = A o e kt Interest rate is the growth rate. Eponential Growth and Deca Models The mathematical model for eponential growth or deca is given b ft = A 0 e kt or A = A 0 e kt. If k>0, the function models the amount, or size, of a growing entit. A 0 is the original amount, or size, of the growing entit at time t = 0, A is the amount at time t, and k is a constant representing the growth rate. If k<0, the function models the amount, or size, of a decaing entit. A 0 is the original amount, or size, of the decaing entit at time t = 0, A is the amount at time t, and k is a constant representing the deca rate. = A 0 e kt k > 0 Increasing A 0 = A 0 e kt k < 0 A 0 t Decreasing t (a) Eponential growth (b) Eponential deca Sometimes we need to use given data to determine k, the rate of growth or deca. After we compute the value of k, we can use the formula A = A 0 e kt to make predictions. This idea is illustrated in our first two eamples. EXAMPLE Modeling the Growth of the U.S. Population The graph in Figure.9 shows the U.S. population, in millions, for five selected ears from 970 through 00. In 970, the U.S. population was 0. million. B 00, it had grown to 94 million. a. Find the eponential growth function that models the data for 970 through 00. b. B which ear will the U.S. population reach 5 million? Solution a. We use the eponential growth model A = A 0 e kt, Population (millions) in which t is the number of ears after 970. This means that 970 corresponds to t = 0. At that time the U.S. population was 0. million, so we substitute 0. for A 0 in the growth model: A = 0.e kt. We are given that 94 million is the population in 00. Because 00 is ears after 970, when t = the value of A is 94. Substituting these numbers into the growth model will enable us to find k, the growth rate. We know that k 7 0 because the problem involves growth Figure Source: Bureau of the Census U.S. Population, Year

50 44 Chapter Eponential and Logarithmic Functions A = 0.e kt 94 = 0.e k # e k = ln e k = lna b k = lna b Use the growth model A A 0 e kt with A When t, A 94. Substitute these numbers into the model. Isolate the eponential factor b dividing both sides b 0.. We also reversed the sides. Take the natural logarithm on both sides. Simplif the left side using ln e. k = lna b L 0.0 Divide both sides b and solve for k. Then use a calculator. The value of k, approimatel 0.0, indicates a growth rate of about.%. This means that the U.S. population is increasing b approimatel.% per ear. We substitute 0.0 for k in the growth model, A = 0.e kt, to obtain the eponential growth function for the U.S. population. It is A = 0.e 0.0t, where t is measured in ears after 970. b. To find the ear in which the U.S. population will reach 5 million, substitute 5 for A in the model from part (a) and solve for t. A = 0.e 0.0t 5 = 0.e 0.0t e 0.0t = 5 0. ln e 0.0t = lna 5 0. b 0.0t = lna 5 0. b This is the model from part (a). Substitute 5 for A. Divide both sides b 0.. We also reversed the sides. Take the natural logarithm on both sides. Simplif on the left using ln e. t = lna 5 0. b 0.0 L 40 Divide both sides b and solve for t. Then use a calculator. Because t represents the number of ears after 970, the model indicates that the U.S. population will reach 5 million b , or in the ear 00. Point In 990, the population of Africa was 64 million and b 000 it had grown to 8 million. a. Use the eponential growth model A = A 0 e kt, in which t is the number of ears after 990, to find the eponential growth function that models the data. b. B which ear will Africa s population reach 000 million, or two billion?

51 Creating an Inaccurate Picture b Leaving Something Out On Monda, October 9, 987, the Dow Jones Industrial Average plunged 508 points, losing.6% of its value. The graph shown on the left, which appeared in a major newspaper following Black Monda (as it was instantl dubbed), creates the impression that the Dow average had been bullish from 97 through 987, increasing throughout this period. The graph creates this inaccurate picture b leaving something out. The graph on the right illustrates that the stock market rose and fell sharpl over these ears. The impressivel smooth curve on the left was obtained b plotting onl three of the data points. B ignoring most of the data, increases and decreases are not accounted for and the actual behavior of the market over the 5 ears leading to Black Monda is inaccuratel conveed. In Eample, we used onl two data values, the population for 970 and the population for 00, to develop a model for U.S. population growth from 970 through 00. B not using data for an other ears, have we created a model that inaccuratel describes both the eisiting data and future population projections given b Section.5 Eponential Growth and Deca; Modeling Data 45 Dow Jones Industrial Average The Dow Jones Industrial Average: Nov., 97 Oct. 6, 987 Growth Using Onl Three Data Points Source: A. K. Dewdne, 00% of Nothing Nov., 97 Oct. 6, 987 Growth and Decline Using All Data the U.S. Census Bureau? Something else to think about: Is an eponential model the best choice for describing U.S. population growth, or might a linear model provide a better description? We return to these issues in Eercises in the eercise set Our net eample involves eponential deca and its use in determining the age of fossils and artifacts. The method is based on considering the percentage of carbon-4 remaining in the fossil or artifact. Carbon-4 decas eponentiall with a half-life of approimatel 575 ears. The half-life of a substance is the time required for half of a given sample to disintegrate. Thus, after 575 ears a given amount of carbon-4 will have decaed to half the original amount. Carbon dating is useful for artifacts or fossils up to 80,000 ears old. Older objects do not have enough carbon-4 left to determine age accuratel. EXAMPLE Carbon-4 Dating: The Dead Sea Scrolls a. Use the fact that after 575 ears a given amount of carbon-4 will have decaed to half the original amount to find the eponential deca model for carbon-4. b. In 947, earthenware jars containing what are known as the Dead Sea Scrolls were found b an Arab Bedouin herdsman. Analsis indicated that the scroll wrappings contained 76% of their original carbon-4. Estimate the age of the Dead Sea Scrolls. Solution a. We begin with the eponential deca model A = A 0 e kt. We know that k 6 0 because the problem involves the deca of carbon-4. After 575 ears t = 575, the amount of carbon-4 present, A, is half the original amount, A 0. A 0 Thus, we can substitute for A in the eponential deca model. This will enable us to find k, the deca rate. A = A 0 e kt Begin with the eponential deca model. A 0 = A 0 e k575 = e575k After 575 ears (t 575), A A 0 (because the amount present, A, is half the original amount, ). A 0 Divide both sides of the equation b A 0.

52 46 Chapter Eponential and Logarithmic Functions Carbon Dating and Artistic Development The artistic communit was electrified b the discover in 995 of spectacular cave paintings in a limestone cavern in France. Carbon dating of the charcoal from the site showed that the images, created b artists of remarkable talent, were 0,000 ears old, making them the oldest cave paintings ever found. The artists seemed to have used the cavern s natural contours to heighten a sense of perspective. The qualit of the painting suggests that the art of earl humans did not mature steadil from primitive to sophisticated in an simple linear fashion. lna b = ln e575k lna b = 575k k = lna b 575 L Take the natural logarithm on both sides. Simplif the right side using ln e. Divide both sides b 575 and solve for k. Substituting for k in the deca model, A = A 0 e kt, the model for carbon-4 is A = A 0 e t. A 0.76A 0. b. In 947, the Dead Sea Scrolls contained 76% of their original carbon-4. To find their age in 947, substitute 0.76A 0 for A in the model from part (a) and solve for t. A = A 0 e t 0.76A 0 = A 0 e t This is the deca model for carbon-4. A, the amount present, is 76% of the original amount, so 0.76 = e t ln 0.76 = ln e t ln 0.76 = t Divide both sides of the equation b A 0. Take the natural logarithm on both sides. Simplif the right side using ln e. t = ln L 68 Divide both sides b and solve for t. The Dead Sea Scrolls are approimatel 68 ears old plus the number of ears between 947 and the current ear. ❷Use logistic growth models. Horizontal asmptote A provides a limit to growth. Increasing rate of growth Original amount at t = 0 Decreasing rate of growth Figure.0 The logistic growth curve has a horizontal asmptote that identifies the limit of the growth of A over time. t Point Strontium-90 is a waste product from nuclear reactors. As a consequence of fallout from atmospheric nuclear tests, we all have a measurable amount of strontium-90 in our bones. a. The half-life of strontium-90 is 8 ears, meaning that after 8 ears a given amount of the substance will have decaed to half the original amount. Find the eponential deca model for strontium-90. b. Suppose that a nuclear accident occurs and releases 60 grams of strontium-90 into the atmosphere. How long will it take for strontium- 90 to deca to a level of 0 grams? Logistic Growth Models From population growth to the spread of an epidemic, nothing on Earth can grow eponentiall indefinitel. Growth is alwas limited. This is shown in Figure.0 b the horizontal asmptote. The logistic growth model is a function used to model situations of this tpe. Logistic Growth Model The mathematical model for limited logistic growth is given b ft = c + ae -bt or A = c + ae -bt, where a, b, and c are constants, with c 7 0 and b 7 0. ae -bt As time increases t : q, the epression in the model approaches 0, and A gets closer and closer to c.this means that = c is a horizontal asmptote for the graph of the function. Thus, the value of A can never eceed c and c represents the limiting size that A can attain.

53 Section.5 Eponential Growth and Deca; Modeling Data 47 Technolog The graph of the logistic growth function for the flu epidemic 0,000 = + 0e -.5 can be obtained using a graphing utilit. We started at 0 and ended at 0. This takes us to week 0. (In Eample, we found that b week 4 approimatel 8,58 people were ill.) We also know that 0,000 is the limiting size, so we took values of up to 0,000. Using a [0, 0, ] b [0, 0,000, 000] viewing rectangle, the graph of the logistic growth function is shown below. EXAMPLE The function Modeling the Spread of the Flu describes the number of people, ft, who have become ill with influenza t weeks after its initial outbreak in a town with 0,000 inhabitants. a. How man people became ill with the flu when the epidemic began? b. How man people were ill b the end of the fourth week? c. What is the limiting size of ft, the population that becomes ill? Solution a. The time at the beginning of the flu epidemic is t = 0. Thus, we can find the number of people who were ill at the beginning of the epidemic b substituting 0 for t. ft = f0 = = 0, L 49 This is the given logistic growth function. When the epidemic began, t 0. Approimatel 49 people were ill when the epidemic began. b. We find the number of people who were ill at the end of the fourth week b substituting 4 for t in the logistic growth function. ft = 0, e -.5t f4 = 0, e -.54 L 8,58 0,000 ft = + 0e -.5t 0, e -.5t 0, e -.50 e.50 e 0 Use the given logistic growth function. To find the number of people ill b the end of week four, let t 4. Use a calculator. Approimatel 8,58 people were ill b the end of the fourth week. Compared with the number of people who were ill initiall, 49, this illustrates the virulence of the epidemic. c c. Recall that in the logistic growth model, ft = the constant c + ae -bt, represents the limiting size that ft can attain. Thus, the number in the numerator, 0,000, is the limiting size of the population that becomes ill. Point In a learning theor project, pschologists discovered that 0.8 ft = + e -0.t is a model for describing the proportion of correct responses, ft, after t learning trials. a. Find the proportion of correct responses prior to learning trials taking place. b. Find the proportion of correct responses after 0 learning trials. c. What is the limiting size of ft, the proportion of correct responses, as continued learning trials take place?

54 48 Chapter Eponential and Logarithmic Functions ❸Use Newton s Law of Modeling Cooling Cooling. Over a period of time, a cup of hot coffee cools to the temperature of the surrounding air. Newton s Law of Cooling, named after Sir Isaac Newton, states that the temperature of a heated object decreases eponentiall over time toward the temperature of the surrounding medium. Stud Tip Newton s Law of Cooling applies to an situation in which an object s temperature is different from that of the surrounding medium, Thus, it can be used to model a heated object cooling to room temperature as well as a frozen object thawing to room temperature. Newton s Law of Cooling The temperature, T, of a heated object at time t is given b T = C + T 0 - Ce kt, where C is the constant temperature of the surrounding medium, is the initial temperature of the heated object, and k is a negative constant that is associated with the cooling object. T 0 EXAMPLE 4 Using Newton s Law of Cooling A cake removed from the oven has a temperature of 0 F. It is left to cool in a room that has a temperature of 70 F. After 0 minutes, the temperature of the cake is 40 F. a. Use Newton s Law of Cooling to find a model for the temperature of the cake, T, after t minutes. b. What is the temperature of the cake after 40 minutes? c. When will the temperature of the cake be 90 F? Solution a. We use Newton s Law of Cooling T = C + T 0 - Ce kt. When the cake is removed from the oven, its temperature is 0 F. This is its initial temperature: T 0 = 0. The constant temperature of the room is 70 F: C = 70. Substitute these values into Newton s Law of Cooling. Thus, the temperature of the cake, T, in degrees Fahrenheit, at time t, in minutes, is T = e kt = e kt. After 0 minutes, the temperature of the cake is 40 F. This means that when t = 0, T = 40. Substituting these numbers into Newton s Law of Cooling will enable us to find k, a negative constant. T = e kt 40 = e k # 0 70 = 40e 0k Use Newton s Law of Cooling from above. When t 0, T 40. Substitute these numbers into the cooling model. Subtract 70 from both sides. e 0k = ln e 0k = ln A B 0k = ln A B k = ln A B 0 L -0.0 Isolate the eponential factor b dividing both sides b 40. We also reversed the sides. Take the natural logarithm on both sides. Simplif the left side using ln e. Divide both sides b 0 and solve for k.

55 Section.5 Section name 49 We substitute -0.0 for k into Newton s Law of Cooling, T = e kt. The temperature of the cake, T, in degrees Fahrenheit, after t minutes is modeled b T = e -0.0t. b. To find the temperature of the cake after 40 minutes, we substitute 40 for t into the cooling model from part (a) and evaluate to find T. T = e L 6 Technolog The graphs illustrate how the temperature of the cake decreases eponentiall over time toward the 70 F room temperature. Temperature ( F) Cake: = e 0.0 Room: = 70 Time (minutes) 0 After 40 minutes, the temperature of the cake will be approimatel 6 F. c. To find when the temperature of the cake will be 90 F, we substitute 90 for T into the cooling model from part (a) and solve for t. T = e -0.0t 90 = e -0.0t 0 = 40e -0.0t e -0.0t = 7 ln e -0.0t = ln A 7B -0.0t = ln A 7B t = ln A 7B -0.0 L 84 This is the cooling model from part (a). Substitute 90 for T. Subtract 70 from both sides. Divide both sides b 40. We also reversed the sides. Take the natural logarithm on both sides. Simplif the left side using ln e. Solve for t b dividing both sides b 0.0. The temperature of the cake will be 90 F after approimatel 84 minutes. ❹ Model data with eponential and logarithmic functions. Point4 An object is heated to 00 C. It is left to cool in a room that has a temperature of 0 C. After 5 minutes, the temperature of the object is 80 C. a. Use Newton s Law of Cooling to find a model for the temperature of the object, T, after t minutes. b. What is the temperature of the object after 0 minutes? c. When will the temperature of the object be 5 C? The Art of Modeling Throughout this chapter, we have been working with models that were given. However, we can create functions that model data b observing patterns in scatter plots. Figure. shows scatter plots for data that are eponential or logarithmic. = ab, a > 0, b > Eponential = ab, a > 0, 0 < b < Eponential = a + b ln, a > 0, b > 0 Logarithmic = a + b ln, a > 0, b < 0 Logarithmic Figure. Scatter plots for eponential or logarithmic models

56 40 Chapter Eponential and Logarithmic Functions EXAMPLE 5 Choosing a Model for Data Figure.(a) shows the percentage of U.S. households with televisions that subscribe to cable television. The data are displaed for five selected ears from 980 through 00. A scatter plot is shown in Figure.(b). What function would be a good choice for modeling the data? Percentage of U.S. Households with TVs with Cable Television Percent Percent Year Year Figure.(a) Source: Nielsen Media Research Figure.(b) Solution Because the data in the scatter plot increase rapidl at first and then begin to level off a bit, the shape suggests that a logarithmic function is a good choice for modeling the data. Point5 Table.5 shows the populations of various cities, in thousands, and the average walking speed, in feet per second, of a person living in the cit. Create a scatter plot for the data. Based on the scatter plot, what function would be a good choice for modeling the data? Table.5 Population and Walking Speed Population Walking Speed (thousands) (feet per second) Source: Mark and Helen Bornstein, The Pace of Life How can we obtain a logarithmic function that models the data for the percentage of U.S. households with cable television shown in Figure.(a)? A graphing utilit can be used to obtain a logarithmic model of the form = a + b ln. Because the domain of the logarithmic function is the set of positive numbers, zero must not be a value for.what does this mean for our cable television data that begin in the ear 980? We must start values of after 0.Thus, we ll assign to represent the number of ears after 979. This gives us the data shown in Table.6. Using the Logarithmic REGression option, we obtain the equation in Figure.. Table.6, Number of, Percentage of U.S. Years after 979 Households with Cable TV (980).6 6 (985) 46. (990) (995) 65.7 (00) 68.9 Figure. A logarithmic model for the data in Table.6

57 Section.5 Eponential Growth and Deca; Modeling Data 4 From Figure., we see that the logarithmic model of the data, with numbers rounded to three decimal places, is = ln. The number r that appears in Figure. is called the correlation coefficient and is a measure of how well the model fits the data. The value of r is such that - r. A positive r means that as the -values increase, so do the -values. A negative r means that as the -values increase, the -values decrease. The closer that r is to or, the better the model fits the data. Because r is approimatel 0.996, the model fits the data ver well. EXAMPLE 6 Choosing a Model for Data Figure.4(a) shows world population, in billions, for seven selected ears from 950 through 00. A scatter plot is shown in Figure.4(b). Suggest two functions that would be good choices for modeling the data. World Population, Population (billions) Population (billions) Year Figure.4(a) Source: U.S. Census Bureau, International Database Year Figure.4(b) Solution Because the data in the scatter plot appear to increase more and more rapidl, the shape suggests that an eponential model might be a good choice. Furthermore, we can probabl draw a line that passes through or near the seven points. Thus, a linear function would also be a good choice for modeling the data. Point6 In 00, 49. million tons of paper were reccled in the United States. Table.7 shows the percentage of all paper reccled for five selected ears from 970 through 00. Create a scatter plot for the data. Based on the scatter plot, what function would be a good choice for modeling the data? Table.7 Year Percentage of All Paper Reccled in the U.S. Percent 970.4% % 990.5% % % Source: The American Forest and Paper Association

58 4 Chapter Eponential and Logarithmic Functions Population (billions) World Population, Year Figure.4(a) (repeated) If we choose to model world population shown in Figure.4(a) with an eponential function, a graphing utilit s Eponential REGression option can be used to obtain the function s equation. With this feature, a graphing utilit fits the data to an eponential model of the form = ab. Although the domain of the eponential function = ab is the set of all real numbers, some graphing utilities onl accept positive values for. What does this mean for our data for world population that starts in the ear 950? We will start values of after 0. Thus, we ll assign to represent the number of ears after 949. This gives us the data shown in Table.8. Using the Eponential REGression option, we obtain the equation in Figure.5. Table.8, Numbers of Years, World Population after 949 (billions) (950).6 (960).0 (970).7 (980) (990) 5. 5 (000) (00) 6. Figure.5 An eponential model for the data in Table.8 From Figure.5, we see that the eponential model of the data for world population ears after 949, with numbers rounded to three decimal places, is Stud Tip Once ou have obtained one or more models for data, ou can use a graphing utilit s TABLE feature to numericall see how well each model describes the data. Enter the models as,, and so on. Create a table, scroll through the table, and compare the table values given b the models to the actual data. ❺ Epress an eponential model in base e. The correlation coefficient, r, is close to, indicating that the model fits the data ver well. When using a graphing utilit to model data, begin with a scatter plot, drawn either b hand or with the graphing utilit, to obtain a general picture for the shape of the data. It might be difficult to determine which model best fits the data linear, logarithmic, eponential, quadratic, or something else. If necessar, use our graphing utilit to fit several models to the data. The best model is the one that ields the value of r, the correlation coefficient, closest to or -. Finding a proper fit for data can be almost as much art as it is mathematics. In this era of technolog, the process of creating models that best fit data is one that involves more decision making than computation. Epressing ab in Base e = Graphing utilities displa eponential models in the form = ab. However, our discussion of eponential growth involved base e. Because of the inverse propert b = e ln b, we can rewrite an model in the form = ab in terms of base e. Epressing an Eponential Model in Base e = ab is equivalent to = ae ln b # EXAMPLE 7 Rewriting the Model for World Population in Base e We have seen that the function = models world population,, in billions, ears after 949. Rewrite the model in terms of base e.

59 Section.5 Eponential Growth and Deca; Modeling Data 4 Solution We use the two equivalent equations shown in the voice balloons to rewrite the model in terms of base e. = ab = ae (ln b) =.557(.07) is equivalent to =.557e (ln.07). Using ln.07 L 0.07, the eponential growth model for world population,, in billions, ears after 949 is =.557e In Eample 7, we can replace with A and with t so that the model has the same letters as those in the eponential growth model A = A 0 e kt. A= A o e kt This is the eponential growth model. A=.557e 0.07t This is the model for world population. The value of k, 0.07, indicates a growth rate of.7%. Although this is an ecellent model for the data, we must be careful about making projections about world population using this growth function. Wh? World population growth rate is now.%, not.7%, so our model will overestimate future populations. Point7 Rewrite = 47.8 in terms of base e. Epress the answer in terms of a natural logarithm and then round to three decimal places. EXERCISE SET.5 Practice Eercises and Application Eercises The eponential models describe the population of the indicated countr, A, in millions, t ears after 00. Use these models to solve Eercises 6. India A=049.7e 0.05t Iraq A=4.7e 0.08t Japan A=7.e 0.00t Russia A=44.5e 0.004t. What was the population of Japan in 00?. What was the population of Iraq in 00?. Which countr has the greatest growth rate? B what percentage is the population of that countr increasing each ear? 4. Which countr has a decreasing population? B what percentage is the population of that countr decreasing each ear? 5. When will India s population be 8 million? 6. When will India s population be 46 million? Population (millions) About the size of New Jerse, Israel has seen its population soar to more than 6 million since it was established.with the help of U.S. aid, the countr now has a diversified econom rivaling those of other developed Western nations. B contrast, the Palestinians, living under Israeli occupation and a corrupt regime, endure bleak conditions. The graphs show that b 050, Palestinians in the West Bank, Gaza Strip, and East Jerusalem will outnumber Israelis. Eercises 7 8, involve the projected growth of these two populations. 9 6 Population of Israel 000: 6,040,000 Projected Year Source: Newsweek Population (millions) 9 6 Palestinian Population in West Bank, Gaza, and East Jerusalem 000:,9,000 Projected Year 7. a. In 000, the population of Israel was approimatel 6.04 million and b 050 it is projected to grow to 0 million. Use the eponential growth model A = A 0 e kt, in which t is the number of ears after 000, to find an eponential growth function that models the data. b. In which ear will Israel s population be 9 million?

60 44 Chapter Eponential and Logarithmic Functions 8. a. In 000, the population of the Palestinians in the West Bank, Gaza Strip, and East Jerusalem was approimatel. million and b 050 it is projected to grow to million. Use the eponential growth model A = A 0 e kt, in which t is the number of ears after 000, to find the eponential growth function that models the data. b. In which ear will the Palestinian population be 9 million? An artifact originall had 6 grams of carbon-4 present. The deca model A = 6e t describes the amount of carbon-4 present after t ears. Use this model to solve Eercises How man grams of carbon-4 will be present in 575 ears? 0. How man grams of carbon-4 will be present in,40 ears?. The half-life of the radioactive element krpton-9 is 0 seconds. If 6 grams of krpton-9 are initiall present, how man grams are present after 0 seconds? 0 seconds? 0 seconds? 40 seconds? 50 seconds?. The half-life of the radioactive element plutonium-9 is 5,000 ears. If 6 grams of plutonium-9 are initiall present, how man grams are present after 5,000 ears? 50,000 ears? 75,000 ears? 00,000 ears? 5,000 ears? Use the eponential deca model for carbon-4, A = A 0 e t, to solve Eercises 4.. Prehistoric cave paintings were discovered in a cave in France. The paint contained 5% of the original carbon-4. Estimate the age of the paintings. 4. Skeletons were found at a construction site in San Francisco in 989. The skeletons contained 88% of the epected amount of carbon-4 found in a living person. In 989, how old were the skeletons? 5. The August 978 issue of National Geographic described the 964 find of bones of a newl discovered dinosaur weighing 70 pounds, measuring 9 feet, with a 6-inch claw on one toe of each hind foot. The age of the dinosaur was estimated using potassium-40 dating of rocks surrounding the bones. a. Potassium-40 decas eponentiall with a half-life of approimatel. billion ears. Use the fact that after. billion ears a given amount of potassium-40 will have decaed to half the original amount to show that the deca model for potassium-40 is given b A = A 0 e -0.59t, where t is in billions of ears. b. Analsis of the rocks surrounding the dinosaur bones indicated that 94.5% of the original amount of potassium- 40 was still present. Let A = 0.945A 0 in the model in part (a) and estimate the age of the bones of the dinosaur. 6. A bird species in danger of etinction has a population that is decreasing eponentiall A = A 0 e kt. Five ears ago the population was at 400 and toda onl 000 of the birds are alive. Once the population drops below 00, the situation will be irreversible. When will this happen? 7. Use the eponential growth model, A = A 0 e kt, to show that the time it takes a population to double (to grow from t = ln k. A 0) is given b 8. Use the eponential growth model, A = A 0 e kt, to show that the time it takes a population to triple (to grow from A 0 to A ) is given b t = ln 0 k. A 0 to Use the formula t = ln that gives the time for a population with k a growth rate k to double to solve Eercises 9 0. Epress each answer to the nearest whole ear. 9. The growth model A = 4e 0.007t describes New Zealand s population, A, in millions, t ears after 00. a. What is New Zealand s growth rate? b. How long will it take New Zealand to double its population? 0. The growth model A = 04.9e 0.07t describes Meico s population, A, in millions, t ears after 00. a. What is Meico s growth rate? b. How long will it take Meico to double its population?. The logistic growth function describes the number of people, ft, who have become ill with influenza t weeks after its initial outbreak in a particular communit. a. How man people became ill with the flu when the epidemic began? b. How man people were ill b the end of the fourth week? c. What is the limiting size of the population that becomes ill? Shown, again, is world population, in billions, for seven selected ears from 950 through 00. Using a graphing utilit s logistic REGression option, we obtain the equation shown on the screen., Numbers of Years, World Population after 949 (billions) (950).6 (960).0 (970).7 (980) (990) 5. 5 (000) (00) 6. We see that a logistic growth model for world population, f, in billions, ears after 949 is f = ft = 00, e -t e Use this function to solve Eercises 6.. How well does the function model the data for 000?. How well does the function model the data for 00? 4. When will world population reach 7 billion? 5. When will world population reach 8 billion?

61 Section.5 Eponential Growth and Deca; Modeling Data According to the model, what is the limiting size of the population that Earth will eventuall sustain? What does this mean in terms of the statement made b the U.S. National Academ of Sciences that 0 billion is the maimum that the world can support with some degree of comfort and individual choice? The logistic growth function 90 P = + 7e -0. models the percentage, P, of Americans who are ears old with some coronar heart disease. Use the function to solve Eercises What percentage of 0-ear-olds have some coronar heart disease? 8. What percentage of 80-ear-olds have some coronar heart disease? 9. At what age is the percentage of some coronar heart disease 50%? 0. At what age is the percentage of some coronar heart disease 70%? Use Newton s Law of Cooling, T = C + T 0 - Ce kt, to solve Eercises 4.. A bottle of juice initiall has a temperature of 70 F. It is left to cool in a refrigerator that has a temperature of 45 F. After 0 minutes, the temperature of the juice is 55 F. a. Use Newton s Law of Cooling to find a model for the temperature of the juice, T, after t minutes. b. What is the temperature of the juice after 5 minutes? c. When will the temperature of the juice be 50 F?. A pizza removed from the oven has a temperature of 450 F. It is left sitting in a room that has a temperature of 70 F. After 5 minutes, the temperature of the pizza is 00 F. a. Use Newton s Law of Cooling to find a model for the temperature of the pizza, T, after t minutes. b. What is the temperature of the pizza after 0 minutes? c. When will the temperature of the pizza be 40 F?. A frozen steak initiall has a temperature of 8 F. It is left to thaw in a room that has a temperature of 75 F. After 0 minutes, the temperature of the steak has risen to 8 F. After how man minutes will the temperature of the steak be 50 F? 4. A frozen steak initiall has a temperature of 4 F. It is left to thaw in a room that has a temperature of 65 F. After 0 minutes, the temperature of the steak has risen to 0 F. After how man minutes will the temperature of the steak be 45 F? Eercises 5 40 present data in the form of tables. For each data set shown b the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an eponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, in Eercise 6, ou will use our graphing utilit to obtain these functions.) 5. Percent of Miscarriages, b Age Woman s Age Percent of Miscarriages 9% Source: Time 7 0% % 7 0% 4 8% 47 5% 6. Number of Countries Connected to the Internet Number of Countries Year Connected to the Internet Source: Medard Gabel, Global Inc., Number of Illegal Immigrants Living in the U.S. Number of Illegal Year Immigrants (millions) Source: U.S. Department of Homeland Securit 8. Number of U.S. Households with Pets Number with Year Pets (millions) Source: American Pet Products Manufacturers Association 9. Alcohol Use b U.S. High School Seniors Percentage Using Alcohol during 0 Das Year Preceding the Surve % % % % % % % Source: U.S. Department of Health and Human Services

62 46 Chapter Eponential and Logarithmic Functions 40. U.S. Vehicle Fatalit Rates Deaths per 00 Million Year Vehicle Miles Traveled Source: National Highwa Traffic Safet Administration In Eercises 4 44, rewrite the equation in terms of base e. Epress the answer in terms of a natural logarithm and then round to three decimal places. 4. = = = = Writing in Mathematics 45. Nigeria has a growth rate of 0.05 or.5%. Describe what this means. 46. How can ou tell whether an eponential model describes eponential growth or eponential deca? 47. Suppose that a population that is growing eponentiall increases from 800,000 people in 00 to,000,000 people in 006. Without showing the details, describe how to obtain the eponential growth function that models the data. 48. What is the half-life of a substance? 49. Describe a difference between eponential growth and logistic growth. 50. Describe the shape of a scatter plot that suggests modeling the data with an eponential function. 5. You take up weightlifting and record the maimum number of pounds ou can lift at the end of each week. You start off with rapid growth in terms of the weight ou can lift from week to week, but then the growth begins to level off. Describe how to obtain a function that models the number of pounds ou can lift at the end of each week. How can ou use this function to predict what might happen if ou continue the sport? 5. Would ou prefer that our salar be modeled eponentiall or logarithmicall? Eplain our answer. 5. One problem with all eponential growth models is that nothing can grow eponentiall forever. Describe factors that might limit the size of a population. Technolog Eercises In Eample on page 4, we used two data points and an eponential function to model the population of the United States from 970 through 00. The data are shown again in the table. Use all five data points to solve Eercises , Number of Years, U.S. Population after 969 (millions) (970) 0. (980) 6.5 (990) 48.7 (000) (00) a. Use our graphing utilit s Eponential REGression option to obtain a model of the form = ab that fits the data. How well does the correlation coefficient, r, indicate that the model fits the data? b. Rewrite the model in terms of base e. B what percentage is the population of the United States increasing each ear? 55. Use our graphing utilit s Logarithmic REGression option to obtain a model of the form = a + b ln that fits the data. How well does the correlation coefficient, r, indicate that the model fits the data? 56. Use our graphing utilit s Linear REGression option to obtain a model of the form = a + b that fits the data. How well does the correlation coefficient, r, indicate that the model fits the data? 57. Use our graphing utilit s Power REGression option to obtain a model of the form = a b that fits the data. How well does the correlation coefficient, r, indicate that the model fits the data? 58. Use the values of r in Eercises to select the two models of best fit. Use each of these models to predict b which ear the U.S. population will reach 5 million. How do these answers compare to the ear we found in Eample, namel 00? If ou obtained different ears, how do ou account for this difference? 59. In Eercises 7 0, ou worked with the logistic growth function 90 P = + 7e -0. which models the percentage, P, of Americans who are ears old with some coronar heart disease. Use our graphing utilit to graph the function in a 0, 00, 04 b 0, 00, 04 viewing rectangle. Describe as specificall as possible what the logistic curve indicates about aging and the percentage of Americans with coronar heart disease. 60. The figure shows the number of people in the United States age 65 and over, with projected figures for the ear 00 and beond. Number of People 65 and Over (millions) U.S. Population Age 65 and Over Source: U.S. Bureau of the Census Year

63 Summar, Review, and Test 47 a. Let represent the number of ears after 899 and let represent the U.S. population age 65 and over, in millions. Use our graphing utilit to find the model that best fits the data in the bar graph. b. Rewrite the model in terms of base e. B what percentage is the 65 and over population increasing each ear? 6. In Eercises 5 40, ou determined the best choice for the kind of function that modeled the data in each table. For each of these eercises that ou worked, use a graphing utilit to find the actual function that best fits the data. Then use the model to make a reasonable prediction for a value that eceeds those shown in the table s first column. Critical Thinking Eercises 6. The eponential growth models describe the population of the indicated countr, A, in millions, t ears after 00. Canada A=.e 0.00t Uganda A=5.6e 0.0t According to these models, which one of the following is true? a. In 00, Uganda s population was ten times that of Canada s. b. In 00, Canada s population eceeded Uganda s b 660,000. c. In 0, Uganda s population will eceed Canada s. d. None of these statements is true. 6. Use Newton s Law of Cooling, T = C + T 0 - Ce -kt, to solve this eercise. At 9:00 A.M., a coroner arrived at the home of a person who had died during the night. The temperature of the room was 70 F, and at the time of death the person had a bod temperature of 98.6 F. The coroner took the bod s temperature at 9:0 A.M., at which time it was 85.6 F, and again at 0:00 A.M., when it was 8.7 F. At what time did the person die? Group Eercises 64. This activit is intended for three or four people who would like to take up weightlifting. Each person in the group should record the maimum number of pounds that he or she can lift at the end of each week for the first 0 consecutive weeks. Use the Logarithmic REGression option of a graphing utilit to obtain a model showing the amount of weight that group members can lift from week through week 0. Graph each of the models in the same viewing rectangle to observe similarities and differences among weight-growth patterns of each member. Use the functions to predict the amount of weight that group members will be able to lift in the future. If the group continues to work out together, check the accurac of these predictions. 65. Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled b eponential or logarithmic functions. Group members should select the two sets of data that are most interesting and relevant. For each set selected, find a model that best fits the data. Each group member should make one prediction based on the model and then discuss a consequence of this prediction. What factors might change the accurac of the prediction? Chapter Summar, Review, and Test Summar DEFINITIONS AND CONCEPTS EXAMPLES. Eponential Functions a. The eponential function with base b is defined b f = b, where b 7 0 and b Z. E., p. 77 b. Characteristics of eponential functions and graphs for 0 6 b 6 and b 7 are shown in the bo on E., p. 78; page 79. E., p. 78 c. Transformations involving eponential functions are summarized in Table. on page 80. E. 4, p. 80; E. 5, p. 8 d. The natural eponential function is f = e. The irrational number e is called the natural base, where E. 6, p. 8 e is the value that a + n e L.78. approaches as n : q. n b e. 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 one of the following formulas:. For n compoundings per ear: A = Pa + r nt n b E. 7, p. 84. For continuous compounding: A = Pe rt

64 48 Chapter Eponential and Logarithmic Functions DEFINITIONS AND CONCEPTS EXAMPLES. Logarithmic Functions a. Definition of the logarithmic function: For 7 0 and b 7 0, b Z, = log is equivalent to b b =. The E., p. 89; function f = log b is the logarithmic function with base b. This function is the inverse function of the E., p. 89; eponential function with base b. E., p. 89 b. Graphs of logarithmic functions for b 7 and 0 6 b 6 are shown in Figure.6 on page 9. Characteristics of the graphs are summarized in the bo that follows the E. 6, p. 9 figure. c. Transformations involving logarithmic functions are summarized in Table.4 on page 9. Figures.7.0, p. 9 d. The domain of a logarithmic function of the form f = log b is the set of all positive real numbers. E. 7, p. 9; The domain of f = log b g consists of all for which g 7 0. E. 0, p. 95 e. Common and natural logarithms: f = log means f = log 0 and is the common logarithmic E. 8, p. 94; function. f = ln means f = log e and is the natural logarithmic function. E. 9, p. 95; E., p. 96 f. Basic Logarithmic Properties Base b Base 0 Base e b>0, b (Common Logarithms) (Natural Logarithms) log b = 0 log = 0 ln = 0 log b b = log 0 = ln e = log b b = log 0 = ln e = b log b = 0 log = e ln = E. 4, p. 90 E. 5, p. 9. Properties of Logarithms a. The Product Rule: log b MN = log b M + log b N E., p. 40 b. The Quotient Rule: log b a M N b = log b M - log b N E., p. 40 c. The Power Rule: log b M p = p log b M E., p. 40 d. The Change-of-Base Propert: The General Introducing Introducing Propert Common Logarithms Natural Logarithms log log log b M = ln M b M = log M b M = log a M log a b log b ln b E. 7, p. 406; E. 8, p. 407 e. Properties for epanding logarithmic epressions are given in the bo on page 40. E. 4, p. 404 f. Properties for condensing logarithmic epressions are given in the bo on page 404. E. 5, p. 404; E. 6, p Eponential and Logarithmic Equations a. An eponential equation is an equation containing a variable in an eponent. Some eponential equations can be solved b epressing each side as a power of the same base: If b M = b N, then M = N. Details are in the bo on page 40. b. The procedure for using natural logarithms to solve eponential equations is given in the bo on page 4. The solution procedure involves isolating the eponential epression and taking the natural logarithm on both sides. c. A logarithmic equation is an equation containing a variable in a logarithmic epression. Some logarithmic equations can be epressed in the form log b M = c. The definition of a logarithm is used to rewrite the equation in eponential form: b c = M. See the bo on page 4. When checking logarithmic equations, reject proposed solutions that produce the logarithm of a negative number or the logarithm of 0 in the original equation. E., p. 40 E., p. 4; E., p. 4; E. 4, p. 4; E. 5, p. 4 E. 6, p. 4; E. 7, p. 45

65 Review Eercises 49 DEFINITIONS AND CONCEPTS EXAMPLES d. Some logarithmic equations can be epressed in the form log b M = log b N. Use the one-to-one propert to E. 8, p. 45 rewrite the equation without logarithms: M = N. See the bo on page Eponential Growth and Deca; Modeling Data a. Eponential growth and deca models are given b A = A 0 e kt in which t represents time, A 0 is the amount E., p. 4; present at t = 0, and A is the amount present at time t. If k 7 0, the model describes growth and k is the E., p. 45 growth rate. If k 6 0, the model describes deca and k is the deca rate. c b. The logistic growth model, given b A =, describes situations in which growth is limited. = c is E., p. 47 -bt + ae a horizontal asmptote for the graph, and growth, A, can never eceed c. c. Newton s Law of Cooling: The temperature, T, of a heated object at time t is given b T = C + T 0 - Ce kt, where C is the constant temperature of the surrounding medium, object, and k is a negative constant. d. Scatter plots for eponential and logarithmic models are shown in Figure. on page 49. When using a graphing utilit to model data, the closer that the correlation coefficient, r, is to - or, the better the model fits the data. E. 5, p. 40; E. 6, p. 4 e. Epressing an Eponential Model in Base e: is equivalent to = ae ln b # = ab. E. 7, p. 4 T 0 is the initial temperature of the heated E. 4, p. 48 Review Eercises. In Eercises 4, the graph of an eponential function is given. Select the function for each graph from the following options:.. f = 4, g = 4 -,. 4. h = -4 -, r = f = and g = - 8. f = A B and g = A 9. f = e and g = e B - Use the compound interest formulas to solve Eercises Suppose that ou have $5000 to invest. Which investment ields the greater return over 5 ears: 5.5% compounded semiannuall or 5.5% compounded monthl?. Suppose that ou have $4,000 to invest. Which investment ields the greater return over 0 ears: 7% compounded monthl or 6.85% compounded continuousl?. A cup of coffee is taken out of a microwave oven and placed in a room. The temperature, T, in degrees Fahrenheit, of the coffee after t minutes is modeled b the function T = e t. The graph of the function is shown in the figure. T 40 In Eercises 5 9, graph f and g in the same rectangular coordinate sstem. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asmptotes. Use the graphs to determine each function s domain and range f = and g = 6. f = and g = T = e t T = t

66 440 Chapter Eponential and Logarithmic Functions Use the graph shown at the bottom of the previous page to answer each of the following questions. a. What was the temperature of the coffee when it was first taken out of the microwave? b. What is a reasonable estimate of the temperature of the coffee after 0 minutes? Use our calculator to verif this estimate. c. What is the limit of the temperature to which the coffee will cool? What does this tell ou about the temperature of the room? In Eercises 5, write each equation in its equivalent eponential form.. = log = log 4 5. log 8 = In Eercises 6 8, write each equation in its equivalent logarithmic form = 6 7. b 4 = = 874 In Eercises 9 9, evaluate each epression without using a calculator. If evaluation is not possible, state the reason. 9. log 0. log log -9. log. log 4. log ln e 5 6. log 7. ln e 8. log log log Graph f = and g = log in the same rectangular coordinate sstem.. Graph f = A B and g = log in the same rectangular coordinate sstem. In Eercises 5, the graph of a logarithmic function is given. Select the function for each graph from the following options: f = log, g = log-, h = log -, r = + log -. In Eercises 6 8, begin b graphing f = log. Then use transformations of this graph to graph the given function. What is the graph s -intercept? What is the vertical asmptote? Use the graphs to determine each function s domain and range. 6. g = log - 7. h = - + log 8. In Eercises 9 40, graph f and g in the same rectangular coordinate sstem. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asmptotes. Use the graphs to determine each function s domain and range. 9. f = log and g = -log f = ln and g = -ln In Eercises 4 4, find the domain of each logarithmic function. 4. f = log f = log - 4. r = log - f = ln -.. In Eercises 44 46, use inverse properties of logarithms to simplif each epression. 44. ln e e ln 46. log On the Richter scale, the magnitude, R, of an earthquake of I intensit I is given b R = log, where I is the intensit of I 0 0 a barel felt zero-level earthquake. If the intensit of an earthquake is 000I 0, what is its magnitude on the Richter scale? 48. Students in a pscholog class took a final eamination. As part of an eperiment to see how much of the course content the remembered over time, the took equivalent forms of the eam in monthl intervals thereafter. The average score, ft, for the group after t months is modeled b the function ft = 76-8 logt +, where 0 t. a. What was the average score when the eam was first given?

67 Review Eercises 44 b. What was the average score after months? 4 months? 6 months? 8 months? one ear? c. Use the results from parts (a) and (b) to graph f. Describe what the shape of the graph indicates in terms of the material retained b the students. 49. The formula. describes the time, t, in weeks, that it takes to achieve master of a portion of a task. In the formula, A represents maimum learning possible, N is the portion of the learning that is to be achieved, and c is a constant used to measure an individual s learning stle. A 50-ear-old man decides to start running as a wa to maintain good health. He feels that the maimum rate he could ever hope to achieve is miles per hour. How man weeks will it take before the man can run 5 miles per hour if c = 0.06 for this person? In Eercises 50 5, use properties of logarithms to epand each logarithmic epression as much as possible. Where possible, evaluate logarithmic epressions without using a calculator. 50. log log 5. ln 64 A In Eercises 54 57, use properties of logarithms to condense each logarithmic epression. Write the epression as a single logarithm whose coefficient is. 54. log b 7 + log b 55. log - log 56. ln + 4 ln 57. In Eercises 58 59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. 58. log 6 7, log In Eercises 60 6, determine whether each equation is true or false. Where possible, show work to support our conclusion. If the statement is false, make the necessar change(s) to produce a true statement. 60. ln ln = log 6. ln e 4 = 4 log = ln e.4 t = c lna A A - N b log 4 64 ln - ln log + 9 log log + = log + In Eercises 64 7, solve each eponential equation. Where necessar, epress the solution set in terms of natural logarithms and use a calculator to obtain a decimal approimation, correct to two decimal places, for the solution = = 5 e = =, e 5 = e = = 7, = 7-7. In Eercises 7 78, solve each logarithmic equation. 7. log 4-5 = e - e - 6 = ln = 5 log + + log - = 4 log - - log + = ln ln + = ln log 4 + = log log The function P = 4.7e -0. models the average atmospheric pressure, P(), in pounds per square inch, at an altitude of miles above sea level. The atmospheric pressure at the peak of Mt. Everest, the world s highest mountain, is 4.6 pounds per square inch. How man miles above sea level, to the nearest tenth of a mile, is the peak of Mt. Everest? 80. The amount of carbon dioide in the atmosphere, measured in parts per million, has been increasing as a result of the burning of oil and coal. The buildup of gases and particles traps heat and raises the planet s temperature, a phenomenon called the greenhouse effect. Carbon dioide accounts for about half of the warming. The function ft = t projects carbon dioide concentration, ft, in parts per million, t ears after 000. Using the projections given b the function, when will the carbon dioide concentration be double the preindustrial level of 80 parts per million? 8. The function W = 0.7 ln models the average walking speed, W, in feet per second, of residents in a cit whose population is thousand. Visitors to New York Cit frequentl feel the are moving too slowl to keep pace with New Yorkers average walking speed of.8 feet per second. What is the population of New York Cit? Round to the nearest thousand. 8. Use the formula for compound interest with n compoundings per ear to solve this problem. How long, to the nearest tenth of a ear, will it take $,500 to grow to $0,000 at 6.5% annual interest compounded quarterl? Use the formula for continuous compounding to solve Eercises How long, to the nearest tenth of a ear, will it take $50,000 to triple in value at 7.5% annual interest compounded continuousl? 84. What interest rate, to the nearest percent, is required for an investment subject to continuous compounding to triple in 5 ears?

68 44 Chapter Eponential and Logarithmic Functions According to the U.S. Bureau of the Census, in 990 there were.4 million residents of Hispanic origin living in the United States. B 000, the number had increased to 5. million. The eponential growth function A =.4e kt describes the U.S. Hispanic population, A, in millions, t ears after 990. a. Find k, correct to three decimal places. b. Use the resulting model to project the Hispanic resident population in 00. c. In which ear will the Hispanic resident population reach 60 million? 86. Use the eponential deca model, A = A 0 e kt, to solve this eercise. The half-life of polonium-0 is 40 das. How long will it take for a sample of this substance to deca to 0% of its original amount? 87. The function 500,000 ft = + 499e -0.9t models the number of people, ft, in a cit who have become ill with influenza t weeks after its initial outbreak. a. How man people became ill with the flu when the epidemic began? b. How man people were ill b the end of the sith week? c. What is the limiting size of ft, the population that becomes ill? 88. Use Newton s Law of Cooling, T = C + T 0 - Ce kt, to solve this eercise. You are served a cup of coffee that has a temperature of 85 F. The room temperature is 65 F. After minutes, the temperature of the coffee is 55 F. a. Write a model for the temperature of the coffee, T, after t minutes. b. When will the temperature of the coffee be 05 F? Eercises present data in the form of tables. For each data set shown b the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an eponential function or a logarithmic function is the better choice for modeling the data. 89. Percentage of the U.S. Population, Ages 5 or Older, with a College Degree Year Percent % 99.4% 995.0% % 00 6.% % 90. Projection of U.S. Jobs Moving Overseas Year Number of Jobs Moving Overseas (millions) Source: Forrester Research, Inc. In Eercises 9 9, rewrite the equation in terms of base e. Epress the answer in terms of a natural logarithm and then round to three decimal places. 9. = = The figure shows world population projections through the ear 50. The data are from the United Nations Famil Planning Program and are based on optimistic or pessimistic epectations for successful control of human population growth. Suppose that ou are interested in modeling these data using eponential, logarithmic, linear, and quadratic functions. Which function would ou use to model each of the projections? Eplain our choices. For the choice corresponding to a quadratic model, would our formula involve one with a positive or negative leading coefficient? Eplain. Population (billions) Year Source: U.N. Projections in World Population Growth 6 billion High projection Medium projection Low projection 0 billion billion 4 billion Source: U.S. Census Bureau

69 Chapter Test 44 Chapter Test. Graph f = and g = + in the same rectangular coordinate sstem.. Graph f = log and g = log - in the same rectangular coordinate sstem.. Write in eponential form: log 5 5 =. 4. Write in logarithmic form: 6 = Find the domain of f = ln -. In Eercises 6 7, use properties of logarithms to epand each logarithmic epression as much as possible. Where possible, evaluate logarithmic epressions without using a calculator. 6. log In Eercises 8 9, write each epression as a single logarithm log + log 9. ln 7 - ln 0. Use a calculator to evaluate log 5 7 to four decimal places. In Eercises 8, solve each equation.. - = = e = e - 6e + 5 = 0 5. log = 6. ln = log + log + 5 = ln ln + = ln 6 log 8 9. On the decibel scale, the loudness of a sound, D, in decibels, is given b D = 0 log I, where I is the intensit of the I 0 sound, in watts per meter, and I 0 is the intensit of a sound barel audible to the human ear. If the intensit of a sound is 0 I 0, what is its loudness in decibels? (Such a sound is potentiall damaging to the ear.) In Eercises 0, simplif each epression. 0. ln e 5. log b b. log 6 Use the compound interest formulas to solve Eercises 5.. Suppose ou have $000 to invest. Which investment ields the greater return over 0 ears: 6.5% compounded semiannuall or 6% compounded continuousl? How much more (to the nearest dollar) is ielded b the better investment? 4. How long, to the nearest tenth of a ear, will it take $4000 to grow to $8000 at 5% annual interest compounded quarterl? 5. What interest rate, to the nearest tenth of a percent, is required for an investment subject to continuous compounding to double in 0 ears? 6. The function A = 8.e -0.00t models the population of German, A, in millions, t ears after 00. a. What was the population of German in 00? b. Is the population of German increasing or decreasing? Eplain. c. In which ear will the population of German be 8.5 million? 7. The 990 population of Europe was 509 million; in 000, it was 79 million. Write the eponential growth function that describes the population of Europe, in millions, t ears after Use the eponential deca model for carbon-4, A = A 0 e t, to solve this eercise. Bones of a prehistoric man were discovered and contained 5% of the original amount of carbon-4. How long ago did the man die? 9. The logistic growth function 40 ft = + 9e -0.65t describes the population, ft, of an endangered species of elk t ears after the were introduced to a nonthreatening habitat. a. How man elk were initiall introduced to the habitat? b. How man elk are epected in the habitat after 0 ears? c. What is the limiting size of the elk population that the habitat will sustain? In Eercises 0, determine whether the values in each table belong to an eponential function, a logarithmic function, a linear function, or a quadratic function Rewrite = in terms of base e. Epress the answer in terms of a natural logarithm and then round to three decimal places