Exponential and Logarithmic Functions

Transcription

1 C H A P T E R 0 Eponential and Logarithmic Functions Water flow (ft /sec) (in thousands) Ma, 95 Record Flood 50,500 ft /sec Water depth (ft) ater is one of the essentials of life, et it is something that most of us take for granted. Among other things, the U.S. Geological Surve (U.S.G.S.) studies freshwater. For over 50 ears the Water W Resources Division of the U.S.G.S. has been gathering basic data about the flow of both freshwater and saltwater from streams and groundwater surfaces. This division collects, compiles, analzes, verifies, organizes, and publishes data gathered from groundwater data collection networks in each of the 50 states, Puerto Rico, and the Trust Territories. Records of stream flow, groundwater levels, and water qualit provide hdrological information needed b local, state, and federal agencies as well as the private sector. There are man instances of the importance of the data collected b the U.S.G.S. For eample, before 987 the Tangipahoa River in Louisiana was used etensivel for swimming and boating. In 987 data gathered b the U.S.G.S. showed that fecal coliform levels in the river eceeded safe levels. Consequentl, Louisiana banned recreational use of the river. Other studies b the Water Resources Division include the results of pollutants on salt marsh environments and the effect that salting highwas in winter has on our drinking water suppl. In Eercises 85 and 86 of Section 0. ou will see how data from the U.S.G.S. is used in a logarithmic function to measure water qualit.

2 50 (0-) Chapter 0 Eponential and Logarithmic Functions 0. EXPONENTIAL FUNCTIONS In this section Definition Domain Graphing Eponential Functions Eponential Equations Applications We have studied functions such as f(), g(), and h(). For these functions the variable is the base. In this section we discuss functions that have a variable as an eponent. These functions are called eponential functions. Definition Some eamples of eponential functions are f(), f(), and f(). helpful hint It is essential that ou have a calculator for this chapter. The most modern calculators are the graphing calculators. The cost a bit more but are worth the price. Eponential Function An eponential function is a function of the form f() a, where a 0 and a. We rule out the base in the definition because f() is the same as the constant function f(). Zero is not used as a base because 0 0 for an positive and nonpositive powers of 0 are undefined. Negative numbers are not used as bases because an epression such as ( ) is not a real number if. E X A M P L E Evaluating eponential functions Let f(), g(), and h(). Find the following. a) f b) f( ) c) g() d) h() a) f 8 b) f( ) 8 c) g() 6 d) h() 9 Note that ( ). For man applications of eponential functions we use base 0 or another base called e. The number e is an irrational number that is approimatel.78. We will see how e is used in compound interest in Eample 0 of this section. Base 0 will be used in the net section. Base 0 is called the common base, and base e is called the natural base.

3 0. Eponential Functions (0-) 5 E X A M P L E calculator close-up Most graphing calculators have kes for the functions 0 and e. Base 0 and base e Let f() 0 and g() e. Find the following and round approimate answers to four decimal places. a) f() b) f(.5) c) g(0) d) g() a) f() b) f(.5) Use the 0 ke on a calculator. c) g(0) e 0 d) g() e 7.89 Use the e ke on a calculator. Domain In the definition of an eponential function no restrictions were placed on the eponent because the domain of an eponential function is the set of all real numbers. So both rational and irrational numbers can be used as the eponent. We have been using rational numbers for eponents since Chapter 7, but we have not et seen an irrational number as an eponent. Even though we do not formall define irrational eponents in this tet, an irrational number such as can be used as an eponent, and ou can evaluate an epression such as b using a calculator. Tr it: Graphing Eponential Functions Even though the domain of an eponential function is the set of all real numbers, we can graph an eponential function b evaluating it for just a few integers. E X A M P L E Eponential functions with base greater than Sketch the graph of each function. a) f() b) g() calculator a) We first make a table of ordered pairs that satisf f() : 0 close-up The graph of f() on a calculator appears to touch the -ais. When drawing this graph b hand, make sure that it does not touch the -ais f() 8 As increases, increases and is alwas positive. Because the domain of the function is (, ), we draw the graph in Fig. 0. as a smooth curve through these points. From the graph we can see that the range is (0, ). 5 f() = 0 FIGURE 0.

4 5 (0-) Chapter 0 Eponential and Logarithmic Functions g() = FIGURE 0. b) Make a table of ordered pairs that satisf g() : 0 g() As increases, increases and is alwas positive. The graph is shown in Fig. 0.. From the graph we see that the range is (0, ). Because e.78, the graph of f() e lies between the graphs of f() and g(), as shown in Fig. 0.. Note that all three functions have the same domain and range and the same -intercept. In general, the function f() a for a has the following characteristics: g() =. The -intercept of the curve is (0, ).. The domain is (, ), and the range is (0, ). FIGURE 0.. The curve approaches the negative -ais but does not touch it.. The -values are increasing as we go from left to right along the curve. 5 (0, ) f() = f() = e E X A M P L E Eponential functions with base between 0 and Graph each function. a) f() b) f() calculator close-up The graph of ( ) is a mirror image of the graph of. 0 a) First make a table of ordered pairs that satisf f() : 0 f() 8 As increases, decreases, getting closer and closer to 0. Draw a smooth curve through these points as shown in Fig f() = 5 f() = FIGURE 0. FIGURE 0.5

5 0. Eponential Functions (0-5) 5 b) Because, we make a table for f() : 0 f() As increases,, or, decreases, getting closer and closer to 0. Draw a smooth curve through these points as shown in Fig Notice the similarities and differences between the eponential function with a and with 0 a. The function f() a for 0 a has the following characteristics:. The -intercept of the curve is (0, ).. The domain is (, ), and the range is (0, ).. The curve approaches the positive -ais but does not touch it.. The -values are decreasing as we go from left to right along the curve. CAUTION An eponential function can be written in more than one form. For eample, f() is the same as f(), or f(). Although eponential functions have the form f() a, other functions that have similar forms are also called eponential functions. Notice how changing the form f() a in the net two eamples changes the shape and location of the graph. E X A M P L E 5 f() = FIGURE 0.6 E X A M P L E 6 = 5 FIGURE 0.7 Changing the shape and location Sketch the graph of f(). Make a table of ordered pairs: 0 f() 7 7 The graph through these points is shown in Fig Changing the shape and location Sketch the graph of. Because ( ), all -coordinates are negative. Make a table of ordered pairs: 0 f() The graph through these points is shown in Fig. 0.7.

6 5 (0-6) Chapter 0 Eponential and Logarithmic Functions Eponential Equations In Chapter 9 we used the horizontal-line test to determine whether a function is oneto-one. Because no horizontal line can cross the graph of an eponential function more than once, eponential functions are one-to-one functions. For an eponential function one-to-one means that if two eponential epressions with the same base are equal, then the eponents are equal. One-to-One Propert of Eponential Functions For a 0 and a, if a m a n, then m n. In the net eample we use the one-to-one propert to solve equations involving eponential functions. E X A M P L E 7 calculator close-up You can see the solution to 8 b graphing and 8. The -coordinate of the point of intersection is the solution to the equation calculator close-up The equation 9 has two solutions because the graphs of 9 and intersect twice. 0 5 Using the one-to-one propert Solve each equation. a) 8 b) 9 c) 8 a) Because 8 is, we can write each side as a power of the same base, : 8 Original equation Write each side as a power of the same base. One-to-one propert Check: 8. The solution set is. b) Because 9, we can write each side as a power of : 9 Original equation ( ) Power of a power rule One-to-one propert Check in the original equation. The solution set is,. c) Because 8 and, we can write each side as a power of : 8 Original equation ( ) Write each side as a power of. Power of a power rule One-to-one propert Check in the original equation. The solution set is.

7 0. Eponential Functions (0-7) 55 The one-to-one propert is also used to find the first coordinate when given the second coordinate of an eponential function. E X A M P L E 8 Finding the -coordinate in an eponential function Let f() and g(). Find if: a) f() b) g() 8 a) Because f() and f(), we can find b solving : 5 Write both sides as a power of the same base. 5 One-to-one propert b) Because g() and g() 8, we can find b solving 8: 8 ( ) Because and 8 Power of a power rule One-to-one propert stud tip Although ou should avoid cramming, there are times when ou have no other choice. In this case concentrate on what is in our class notes and the homework assignments. Tr to work one or two problems of each tpe. Instructors often ask some relativel eas questions on a test to see if ou have understood the major ideas. Applications Eponential functions are used to describe phenomena such as population growth, radioactive deca, and compound interest. Here we discuss compound interest. If an investment is earning compound interest, then interest is periodicall paid into the account and the interest that is paid also earns interest. If a bank pas 6% compounded quarterl on an account, then the interest is computed four times per ear (ever months) at.5% (one-quarter of 6%). Suppose an account has $5000 in it at the beginning of a quarter. We can appl the simple interest formula A P Prt, with r 6% and t, to find how much is in the account at the end of the first quarter. A P Prt P( rt) Factor Substitute. 5000(.05) $5075 To repeat this computation for another quarter, we multipl $5075 b.05. If A represents the amount in the account at the end of n quarters, we can write A as an eponential function of n: A $5000(.05) n

8 56 (0-8) Chapter 0 Eponential and Logarithmic Functions In general, the amount A is given b the following formula. Compound Interest Formula If P represents the principal, i the interest rate per period, n the number of periods, and A the amount at the end of n periods, then A P( i) n. E X A M P L E 9 calculator close-up Graph 50(.0) to see the growth of the $50 deposit in Eample 9 over time. After 60 months it is worth $, , helpful hint Compare Eamples 9 and 0 to see the difference between compounded monthl and compounded continuousl. Although there is not much difference to an individual investor, there could be a large difference to the bank. Rework Eamples 9 and 0 using $50 million as the deposit. Compound interest formula If $50 is deposited in an account paing % compounded monthl, then how much is in the account at the end of 6 ears and 6 months? Interest is paid times per ear, so the account earns of %, or % each month, for 78 months. So i 0.0, n 78, and P $50: A P( i) n A $50(.0) 78 $ If we shorten the length of the time period (earl, quarterl, monthl, dail, hourl, etc.), the number of periods n increases while the interest rate for the period decreases. As n increases, the amount A also increases but will not eceed a certain amount. That certain amount is the amount obtained from continuous compounding of the interest. It is shown in more advanced courses that the following formula gives the amount when interest is compounded continuousl. Continuous-Compounding Formula If P is the principal or beginning balance, r is the annual percentage rate compounded continuousl, t is the time in ears, and A is the amount or ending balance, then A Pe rt. CAUTION The value of t in the continuous-compounding formula must be in ears. For eample, if the time is ear and months, then t.5 ears. If the time is ears and 5 das, then t ears. E X A M P L E 0 Continuous-compounding formula If $50 is deposited in an account paing % compounded continuousl, then how much is in the account after 6 ears and 6 months?

9 0. Eponential Functions (0-9) 57 Use r %, t 6.5 ears, and P $50 in the formula for compounding interest continuousl: A Pe rt 50e (0.)(6.5) 50e 0.78 $76.5 Use the e ke on a scientific calculator. Note that compounding continuousl amounts to a few dollars more than compounding monthl did in Eample 9. calculator close-up Graph 50e 0. to see the growth of the $50 deposit in Eample 0 over time. After 0 ears it is worth $, , M A T H A T W O R K Neal Driscoll, a geophsicist at the Lamont-Dohert Earth Observator of Columbia Universit, eplores both the ocean and the continents to understand the processes that shape the earth. What he finds fascinating is the interaction between the ocean and the land not just at the shoreline, but underneath the sea as well. To get a preliminar picture of the ocean floor, Dr. Driscoll spent part of the past summer working with the U.S.G.S., studing the effects of storms on beaches and underwater landscapes. The results of these studies can be used as a baseline to provide help to coastal planners who are building waterfront homes. Other information obtained can be used to direct transporters of dredged material to places where the material is least likel to affect plant, fish, and human life. The most recent stud found man different tpes of ocean floor, ranging from sand and mud to large tracts of algae. Imaging the seafloor is a difficult problem. It can be a costl venture, and there are numerous logistical problems. Recentl developed technolog, such as towable undersea cameras and satellite position sstems, has made the task easier. Dr. Driscoll and his team use this new technolog and sound reflection to gather data about how the sediment on the ocean floor changes in response to storm events. This research is funded b the Office of Naval Research (ONR). In Eercise 8 of the Making Connections eercises ou will see how a geophsicist uses sound to measure the depth of the ocean. GEOPHYSICIST

10 58 (0-0) Chapter 0 Eponential and Logarithmic Functions WARM-UPS True or false? Eplain our answer.. If f(), then f. False. If f(), then f( ). True. The function f() is an eponential function. False. The functions f() and g() have the same graph. True 5. The function f() is invertible. True 6. The graph of has an -intercept. False 7. The -intercept for f() e is (0, ). True 8. The epression is undefined. False 9. The functions f() and g() have the same graph. True 0. If $500 earns 6% compounded monthl, then at the end of ears the investment is worth 500(.005) dollars. False 0. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is an eponential function? An eponential function has the form f() a where a 0 and a.. What is the domain of ever eponential function? The domain of an eponential function is all real numbers.. What are the two most popular bases? The two most popular bases are e and 0.. What is the one-to-one propert of eponential functions? The one-to-one propert states that if a m a n, then m n. 5. What is the compound interest formula? The compound interest formula is A P( i) n. 6. What does compounded continuousl mean? When mone is compounded continuousl, we use the formula A Pe rt. Let f(), g(), and h(). Find the following. See Eample. 7. f() 6 8. f( ) 9. f. h() 00. h(.) j().78. j(.5).5 5. j( ) j(0) Sketch the graph of each function. See Eamples and. 7. f() 8. g() 5 9. h() 0. i() 5 0. f 8. g( ). g() 9. g(0). g( ) 9 5. h(0). 0. (0.) 6. h() 8 7. h( ) 8. h( ) 6 Let h() 0 and j() e. Find the following. Use a calculator as necessar and round approimate answers to four decimal places. See Eample. 9. h(0) 0. h( ) 0.

11 0. Eponential Functions (0-) 59 Sketch the graph of each function. See Eamples 5 and Solve each equation. See Eample f() 6. k() , , 9 Let f(), g(), and h(). Find in each case. See Eample f() 60. f() 7. g() 8. A() 0 6. f() 6. f() 0 6. g() 9 6. g() g() g() 67. h() h() 9. f() e 0. g() e 69. h() 70. h() 5 8. H() 0. s() ( ) Solve each problem. See Eample Compounding quarterl. If $6,000 is deposited in an account paing 5% compounded quarterl, then what amount will be in the account after 0 ears? $9, Compounding quarterl. If $00 is deposited in an account paing 0% compounded quarterl, then what amount will be in the account after 7 ears? $ Outstanding performance. The top stock fund over 0 ears was Fidelit Select-Home Finance because it. P 5000(.05) t. d t Amount (in thousands of dollars) $0,000 Invested Number of ears after 988 FIGURE FOR EXERCISE 7

12 50 (0-) Chapter 0 Eponential and Logarithmic Functions returned an average of 7.6% annuall for 0 ears (Mone s 998 Guide to Mutual Funds, a) How much was an investment of $0,000 in this fund in 988 worth in 998? $,.6 b) Use the accompaning graph to estimate the ear in which the $0,000 investment was worth $75, Second place. The Kaufman fund was the second best fund over 0 ears with an average annual return of 6.5% (Mone s 998 Guide to Mutual Funds, How much was an investment of $0,000 in this fund in 988 worth in 998? $0, Depreciating knowledge. The value of a certain tetbook seems to decrease according to the formula V 5 0.9t, where V is the value in dollars and t is the age of the book in ears. What is the book worth when it is new? What is it worth when it is ears old? $5, $ Mosquito abatement. In a Minnesota swamp in the springtime the number of mosquitoes per acre appears to grow according to the formula N 0 0.t, where t is the number of das since the last frost. What is the size of the mosquito population at times t 0, t 0, and t 0?,000, 0,000, 00,000 In Eercises 77 8, solve each problem. See Eample Compounding continuousl. If $500 is deposited in an account paing 7% compounded continuousl, then how much will be in the account after ears? $ Compounding continuousl. If $7,000 is deposited in an account paing 8% compounded continuousl, then what will it amount to after ears? $9, One ear s interest. How much interest will be earned the first ear on $80,000 on deposit in an account paing 7.5% compounded continuousl? $6, Partial ear. If $7,500 is deposited in an account paing 6.75% compounded continuousl, then how much will be in the account after 5 ears and 5 das? $0, Radioactive deca. The number of grams of a certain radioactive substance present at time t is given b the formula A 00 e 0.06t, where t is the number of ears. Find the amount present at time t 0. Find the amount present Amount (grams) after 0 ears. Use the accompaning graph to estimate the number of ears that it takes for one-half of the substance to deca. Will the substance ever deca completel? 00 grams, 90. grams, ears, no A A 00 e 0.06t 0 0 Years FIGURE FOR EXERCISE 8 8. Population growth. The population of a certain countr appears to be growing according to the formula P 0 e 0.t, where P is the population in millions and t is the number of ears since 980. What was the population in 980? What will the population be in the ear 000? 0 million, 7.8 million GETTING MORE INVOLVED 8. Eploration. An approimate value for e can be found b adding the terms in the following infinite sum:... Use a calculator to find the sum of the first four terms. Find the difference between the sum of the first four terms and e. (For e, use all of the digits that our calculator gives for e.) What is the difference between e and the sum of the first eight terms? , 0.056, GRAPHING CALCULATOR EXERCISES 8. Graph, e, and on the same coordinate sstem. Which point do all three graphs have in common? (0, ) 85. Graph,, and on the same coordinate sstem. What can ou sa about the graph of k for an real number k? The graph of k lies k units to the right of when k 0 and k units to the left of when k 0. t