SECTION 51 Exponential Functions


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1 354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational functions. The general class of functions defined b means of the algebraic operations of addition, subtraction, multiplication, division, and the taking of powers and roots on variables and constants are called algebraic functions. In this chapter we define and investigate the properties of two new and important tpes of functions called eponential functions and logarithmic functions. These functions are not algebraic, but are members of another class of functions called transcendental functions. The eponential functions and logarithmic functions are used in describing and solving a wide variet of realworld problems, including growth of populations of people, animals, and bacteria; radioactive deca; growth of mone at compound interest; absorption of light as it passes through air, water, or glass; and magnitudes of sounds and earthquakes. We consider applications in these areas plus man more in the sections that follow. SECTION 5 Eponential Functions Eponential Functions Basic Eponential Graphs Additional Eponential Properties Applications In this section we define eponential functions, look at some of their important properties including graphs and consider several significant applications. Eponential Functions Let s start b noting that the functions f and g given b f() and g() are not the same function. Whether a variable appears as an eponent with a constant base or as a base with a constant eponent makes a big difference. The function g is a quadratic function, which we have alread discussed. The function f is a new tpe of function called an eponential function. Man students, if asked to graph an eponential function such as f(), would not hesitate at all. The would likel make up a table b assigning integers to, plot the resulting points, and then join these points with a smooth curve, as shown in Figure.
2 5 Eponential Functions 355 FIGURE f (). 0 f() f() The onl catch is that we have not defined for all real numbers. We know what 5, 3, /3, 3/5,.4, and 3.5 mean because we have defined p for an rational number p, but what does mean? The question is not eas to answer at this time. In fact, a precise definition of must wait for more advanced courses, where we can show that, if b is a positive real number and is an real number, then b names a real number, and the graph of f() is as indicated in Figure. We also can show that for irrational, b can be approimated as closel as we like b using rational number approimations for. Since , for eample, the sequence.4,.4,.44,... approimates, and as we use more decimal places, the approimation improves. DEFINITION Eponential Function The equation f() b b 0, b defines an eponential function for each different constant b, called the base. The independent variable ma assume an real value. Thus, the domain of f is the set of all real numbers, and it can be shown that the range of f is the set of all positive real numbers. We require the base b to be positive to avoid imaginar numbers such as ( ) /.
3 356 5 Eponential and Logarithmic Functions Basic Eponential Graphs EXPLOREDISCUSS Compare the graphs of f() 3 and g() b plotting both functions on the same coordinate sstem. Find all points of intersection of the graphs. For which values of is the graph of f above the graph of g? Below the graph of g? Are the graphs of f and g close together as? As? Discuss. It is useful to compare the graphs of and ( ) b plotting both on the same coordinate sstem, as shown in Figure (a). The graph of f() b b [Fig. (b)] looks ver much like the graph of the particular case, and the graph of f() b 0 b [Fig. (b)] FIGURE Basic eponential graphs b 0 b b b 4 4 DOMAIN (, ) RANGE (0, ) (a) looks ver much like the graph of ( ). Note in both cases that the ais is a horizontal asmptote for the graph. The graphs in Figure suggest the following important general properties of eponential functions, which we state without proof: (b) Basic Properties of the Graph of f() b, b 0, b. All graphs pass through the point (0, ). b 0 for an permissible base b.. All graphs are continuous, with no holes or jumps. 3. The ais is a horizontal asmptote. 4. If b, then b increases as increases. 5. If 0 b, then b decreases as increases. 6. The function f is onetoone.
4 5 Eponential Functions 357 Propert 6 implies that an eponential function has an inverse, called a logarithmic function, which we will discuss in Section 53. A calculator ma be used to create an accurate table of values from which the graph of an eponential function is drawn. Eample illustrates the process. (Of course, we ma bpass the creation of a table of values with a graphing utilit, which graphs the function directl.) EXAMPLE Graphing Multiples of Eponential Functions Use integer values of from 3 to 3 to construct a table of values for (4 ), and then graph this function. Solution Use a calculator to create the table of values shown below. Then plot the points, and join these points with a smooth curve (see Fig. 3). FIGURE 3 (4 ) Matched Problem Repeat Eample for ( 4 ) (4 ). Additional Eponential Properties Eponential functions, whose domains include irrational numbers, obe the familiar laws of eponents we discussed earlier for rational eponents. We summarize these eponent laws here and add two other important and useful properties.
5 358 5 Eponential and Logarithmic Functions Eponential Function Properties For a and b positive, a, b, and and real:. Eponent laws: a a a (a ) a (ab) a b a b a b a a a a a if and onl if. If , then 4 4, and. 3. For 0, then a b if and onl if a b. If a 4 3 4, then a 3. EXAMPLE Using Eponential Function Properties Solve for. Solution Epress both sides in terms of the same base, and use propert to equate eponents ( ) Epress 4 and 8 as powers of. (a ) a Propert 9 Check 4 (9/) 3 4 3/ ( 4) Matched Problem Solve 7 9 for. Applications We now consider three applications that utilize eponential functions in their analsis: population growth, radioactive deca, and compound interest. Population growth and compound interest are eamples of eponential growth, while radioactive deca is an eample of negative eponential growth. Our first eample involves the growth of populations, such as people, animals, insects, and bacteria. Populations tend to grow eponentiall and at different rates. A convenient and easil understood measure of growth rate is the doubling time that is, the time it takes for a population to double. Over short periods of time the doubling time growth model is often used to model population growth: P P 0 t/d
6 5 Eponential Functions 359 where P Population at time t P 0 Population at time t 0 d Doubling time Note that when t d, P P 0 d/d P 0 and the population is double the original, as it should be. We use this model to solve a population growth problem in Eample 3. EXAMPLE 3 Population Growth Meico has a population of around 00 million people, and it is estimated that the population will double in ears. If population growth continues at the same rate, what will be the population: (A) 5 ears from now? (B) 30 ears from now? Calculate answers to 3 significant digits. Solutions We use the doubling time growth model: 500 P (millions) P P 0 t/d Substituting P 0 00 and d, we obtain 400 P 00( t/ ) See Figure (A) Find P when t 5 ears: 00 P 00( 5/ ) Years FIGURE 4 P 00( t/ ) 50 t 64 million people (B) Find P when t 30 ears: Use a calculator. P 00( 30/ ) 69 million people Use a calculator. Matched Problem 3 The bacterium Escherichia coli (E. coli) is found naturall in the intestines of man mammals. In a particular laborator eperiment, the doubling time for E. coli is found to be 5 minutes. If the eperiment starts with a population of,000 E. coli and there is no change in the doubling time, how man bacteria will be present: (A) In 0 minutes? (B) In 5 hours? Write answers to 3 significant digits.
7 360 5 Eponential and Logarithmic Functions EXPLOREDISCUSS The doubling time growth model would not be epected to give accurate results over long periods of time. According to the doubling time growth model of Eample 3, what was the population of Meico 500 ears ago at the height of Aztec civilization? What will the population of Meico be 00 ears from now? Eplain wh these results are unrealistic. Discuss factors that affect human populations that are not taken into account b the doubling time growth model. Our second application involves radioactive deca, which is often referred to as negative growth. Radioactive materials are used etensivel in medical diagnosis and therap, as power sources in satellites, and as power sources in man countries. If we start with an amount A 0 of a particular radioactive isotope, the amount declines eponentiall in time. The rate of deca varies from isotope to isotope. A convenient and easil understood measure of the rate of deca is the halflife of the isotope that is, the time it takes for half of a particular material to deca. In this section we use the following halflife deca model: A A 0 ( /h )t t /h A 0 where A Amount at time t A 0 Amount at time t 0 h Halflife Note that when t h, A A 0 h /h A 0 and the amount of isotope is half the original amount, as it should be. A 0 EXAMPLE 4 Radioactive Deca The radioactive isotope gallium 67 ( 67 Ga), used in the diagnosis of malignant tumors, has a biological halflife of 46.5 hours. If we start with 00 milligrams of the isotope, how man milligrams will be left after: (A) 4 hours? (B) week? Compute answers to 3 significant digits. Solutions We use the halflife deca model: Using A 0 00 and h 46.5, we obtain A A 0 ( ) t/h A 0 t/h A 00( t /46.5 ) See Figure 5.
8 5 Eponential Functions A (milligrams) (A) Find A when t 4 hours: A 00( 4/46.5 ) 69.9 milligrams Use a calculator. (B) Find A when t 68 hours ( week 68 hours): Hours FIGURE 5 A 00( t/46.5 ). t A 00( 68/46.5 ) 8.7 milligrams Use a calculator. Matched Problem 4 Radioactive gold 98 ( 98 Au), used in imaging the structure of the liver, has a halflife of.67 das. If we start with 50 milligrams of the isotope, how man milligrams will be left after: (A) da? (B) week? Compute answers to 3 significant digits. Our third application deals with the growth of mone at compound interest. This topic is important to most people and is fundamental to man topics in the mathematics of finance. The fee paid to use another s mone is called interest. It is usuall computed as a percent, called the interest rate, of the principal over a given period of time. If, at the end of a pament period, the interest due is reinvested at the same rate, then the interest earned as well as the principal will earn interest during the net pament period. Interest paid on interest reinvested is called compound interest. Suppose ou deposit $,000 in a savings and loan that pas 8% compounded semiannuall. How much will the savings and loan owe ou at the end of ears? Compounded semiannuall means that interest is paid to our account at the end of each 6month period, and the interest will in turn earn interest. The interest rate per period is the annual rate, 8% 0.08, divided b the number of compounding periods per ear,. If we let A, A, A 3, and A 4 represent the new amounts due at the end of the first, second, third, and fourth periods, respectivel, then A $,000 $, $,000( 0.04) A A ( 0.04) [$,000( 0.04)]( 0.04) $,000( 0.04) A 3 A ( 0.04) [$,000( 0.04) ]( 0.04) $,000( 0.04) 3 P r n P r n P r n 3
9 36 5 Eponential and Logarithmic Functions A 4 A 3 ( 0.04) [$,000( 0.04) 3 ]( 0.04) $,000( 0.04) 4 P r n 4 What do ou think the savings and loan will owe ou at the end of 6 ears? If ou guessed A $,000( 0.04) ou have observed a pattern that is generalized in the following compound interest formula: Compound Interest If a principal P is invested at an annual rate r compounded n times a ear, then the amount A in the account at the end of t ears is given b A P r n nt The annual rate r is epressed in decimal form. Since the principal P represents the initial amount in the account and A represents the amount in the account t ears later, we also call P the present value of the account and A the future value of the account. EXAMPLE 5 Compound Interest 5,000 A (dollars) Solution If ou deposit $5,000 in an account paing 9% compounded dail, how much will ou have in the account in 5 ears? Compute the answer to the nearest cent. We use the compound interest formula as follows: A P r n nt 0,000 5, (365)(5) $7,84. Use a calculator. 5,000 The graph of FIGURE 6 5 Years 0 t is shown in Figure 6. A 5, t
10 5 Eponential Functions 363 Matched Problem 5 If $,000 is invested in an account paing 0% compounded monthl, how much will be in the account at the end of 0 ears? Compute the answer to the nearest cent. EXAMPLE 6 Visualizing Investments with a Graphing Utilit Use a graphing utilit to compare the growth of an investment of $,000 at 0% compounded monthl with an investment of $,000 at 5% compounded monthl. When do the two investments have the same value? Solution 0 We use the compound interest formula to epress the future value of the first investment b,000( 0.0/), and the future value of the second investment b,000( 0.05/), where is time in ears. We graph both functions and use the intersection routine of the graphing utilit to conclude that the investments have the same value when 4 ears, as shown in Figure 7. After that time the $,000 investment has the greater value. FIGURE 7 0 Matched Problem 6 Use a graphing utilit to determine when an investment of $5,000 at 6% compounded quarterl has the same value as an investment of $4,000 at 0% compounded dail. Answers to Matched Problems. (4 ) (A),30 (B) 4,00, (A) 43.9 mg (B) 8. mg 5. $, ears, 6 months
11 364 5 Eponential and Logarithmic Functions EXERCISE 5 A In Problems 0, construct a table of values for integer values of over the indicated interval and then graph the function.. 3 ; [ 3, 3]. 5 ; [, ] 3. ( 3 ) 3 ; [ 3, 3] 4. ( 5 ) 5 ; [, ] 5. g() 3 ; [ 3, 3] 6. f() 5 ; [, ] 7. h() 5(3 ); [ 3, 3] 8. f() 4(5 ); [, ] ; [ 6, 0] ; [ 4, 0] In Problems, simplif (4 ) 3 6. ( 3 ) 4 7. (0 ) (0 ) a b 3 c a 3 b c 3 B In Problems 3 34, solve for ( ) ( ) ,000 4 a 4 b 3 c a 6 b c Find all real numbers a such that a a. Eplain wh this does not violate the second eponential function propert in the bo on page Find real numbers a and b such that a b but a 4 b 4. Eplain wh this does not violate the third eponential function propert in the bo on page 358. Graph each function in Problems b constructing a table of values. * Check Problems with a graphing utilit. 37. G(t) 3 t/ f(t) t/0 39. (3 / ) *Please note that use of graphing utilit is not required to complete these eercises. Checking them with a g.u. is optional ( ) 4. g() 4. f() 43.,000(.08) (.03) C In Problems 47 50, simplif. 47. (6 6 )(6 6 ) 48. (3 3 )(3 3 ) 49. (6 6 ) (6 6 ) 50. (3 3 ) (3 3 ) Graph each function in Problems 5 54 b constructing a table of values. Check Problems 5 54 with a graphing utilit. 5. m() (3 ) 5. h() ( ) f() 54. g() In Problems 55 58: (A) Approimate the real zeros of each function to two decimal places. (B) Investigate the behavior of each function as and and find an horizontal asmptotes. 55. f() f() f() f() 8 APPLICATIONS Gaming. A person bets on red and black on a roulette wheel using a Martingale strateg. That is, a $ bet is placed on red, and the bet is doubled each time until a win occurs. The process is then repeated. If black occurs n times in a row, then L n dollars is lost on the nth bet. Graph this function for n 0. Even though the function is defined onl for positive integers, points on this tpe of graph are usuall jointed with a smooth curve as a visual aid. 60. Bacterial Growth. If bacteria in a certain culture double ever hour, write an equation that gives the number of bacteria N in the culture after t hours, assuming the culture has 00 bacteria at the start. Graph the equation for 0 t Population Growth. Because of its short life span and frequent breeding, the fruit fl Drosophila is used in some genetic studies. Ramond Pearl of Johns Hopkins Universit,
12 5 The Eponential Function with Base e 365 for eample, studied 300 successive generations of descendants of a single pair of Drosophila flies. In a laborator situation with ample food suppl and space, the doubling time for a particular population is.4 das. If we start with 5 male and 5 female flies, how man flies should we epect to have in: (A) week? (B) weeks? 6. Population Growth. If Kena has a population of about 30,000,000 people and a doubling time of 9 ears and if the growth continues at the same rate, find the population in: (A) 0 ears (B) 30 ears Compute answers to significant digits. 63. Insecticides. The use of the insecticide DDT is no longer allowed in man countries because of its longterm adverse effects. If a farmer uses 5 pounds of active DDT, assuming its halflife is ears, how much will still be active after: (A) 5 ears? (B) 0 ears? Compute answers to significant digits. 64. Radioactive Tracers. The radioactive isotope technetium 99m ( 99m Tc) is used in imaging the brain. The isotope has a halflife of 6 hours. If milligrams are used, how much will be present after: (A) 3 hours? (B) 4 hours? Compute answers to 3 significant digits. 65. Finance. Suppose $4,000 is invested at % compounded weekl. How much mone will be in the account in: (A) ear? (B) 0 ears? Compute answers to the nearest cent. 66. Finance. Suppose $,500 is invested at 7% compounded quarterl. How much mone will be in the account in: 3 (A) 4 ear? (B) 5 ears? Compute answers to the nearest cent. 67. Finance. A couple just had a new child. How much should the invest now at 8.5% compounded dail in order to have $40,000 for the child s education 7 ears from now? Compute the answer to the nearest dollar. 68. Finance. A person wishes to have $5,000 cash for a new car 5 ears from now. How much should be placed in an account now if the account pas 9.75% compounded weekl? Compute the answer to the nearest dollar. 69. Finance. Will an investment of $0,000 at 8.9% compounded dail ever be worth more at the end of a quarter than an investment of $0,000 at 9% compounded quarterl? Eplain. 70. Finance. A sum of $5,000 is invested at 3% compounded semiannuall. Suppose that a second investment of $5,000 is made at interest rate r compounded dail. For which values of r, to the nearest tenth of a percent, is the second investment better than the first? Discuss. SECTION 5 The Eponential Function with Base e Base e Eponential Function Growth and Deca Applications Revisited Continuous Compound Interest A Comparison of Eponential Growth Phenomena Until now the number has probabl been the most important irrational number ou have encountered. In this section we will introduce another irrational number, e, that is just as important in mathematics and its applications. Base e Eponential Function The following epression is important to the stud of calculus and, as we will see later in this section, also is closel related to the compound interest formula discussed in the preceding section: m m
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