LOGARITHMIC FUNCTIONS AND THEIR APPLICATIONS

Transcription

1 . Logarithmic Functions and Their Applications ( 3) 657 In this section. LOGARITHMIC FUNCTIONS AND THEIR APPLICATIONS In Section. you learned that eponential functions are one-to-one functions. Because they are one-to-one functions, they have inverse functions. In this section we study the inverses of the eponential functions. Definition Domain and Range Graphing Logarithmic Functions Logarithmic Equations Applications Definition We define log a () as the eponent that is used on the base a to obtain. Read log a () as the base a logarithm of. The epression log a () is called a logarithm. Because 3 8, the eponent is 3 and log (8) 3. Because 5 5, the eponent is and log 5 (5). Because 5 3, the eponent is 5andlog 3 5. So the logarithmic equation y log a () is equivalent to the eponential equation a y. log a () For any a 0anda, y log a () if and only if a y. E X A M P L E Using the definition of logarithm Write each logarithmic equation as an eponential equation and each eponential equation as a logarithmic equation. a) log 5 (5) 3 b) 6 log 4 () c) m 8 d) 7 3 z Domain of f Range of f f() = 5 3 f () = log () a) The base-5 logarithm of 5 equals 3 means that 3 is the eponent on 5 that produces 5. So b) The equation 6 log 4 () is equivalent to 4 6 by the definition of logarithm. c) The equation m 8 is equivalent to log (8) m. d) The equation 7 3 z is equivalent to log 3 (7) z. Range of f Domain of f FIGURE.8 The inverse of the base-a eponential function f() a is the base-a logarithmic function f () log a (). For eample, f() and f () log () are inverse functions as shown in Fig..8. Each function undoes the other. f(5) 5 3 and g(3) log (3) 5. To evaluate logarithmic functions remember that a logarithm is an eponent: log a () is the eponent that is used on the base a to obtain. E X A M P L E Finding logarithms Evaluate each logarithm. a) log 5 (5) b) log 8 c) log (4) d) log 0 (0.00) e) log 9 (3)

2 658 ( 4) Chapter Eponential and Logarithmic Functions helpful hint When we write C(),we may think of C as a variable and write C, or we may think of C as the name of a function, the cost function. In y log a () we are thinking of log a only as the name of the function that pairs an -value with a y-value. E X A M P L E 3 calculator close-up A graphing calculator has keys for the common logarithm (LOG) and the natural logarithm (LN). a) The number log 5 (5) is the eponent that is used on the base 5 to obtain 5. Because 5 5, we have log 5 (5). b) The number log 8 is the power of that gives us 8. Because 8 3, we have log 8 3. c) The number log (4) is the power of that produces 4. Because 4, we have log (4). d) Because , we have log 0 (0.00) 3. e) Because 9 3, we have log 9 (3). There are two bases for logarithms that are used more frequently than the others: They are 0 and e. The base-0 logarithm is called the common logarithm and is usually written as log(). The base-e logarithm is called the natural logarithm and is usually written as ln(). Most scientific calculators have function keys for log() andln(). The simplest way to obtain a common or natural logarithm is to use a scientific calculator. However, a table of common logarithms can be found in Appendi C of this tet. In the net eample we find natural and common logarithms of certain numbers without a calculator or a table. Finding common and natural logarithms Evaluate each logarithm. a) log(000) b) ln(e) c) log 0 a) Because , we have log(000) 3. b) Because e e, we have ln(e). c) Because 0 0, we have log 0. Domain and Range The domain of the eponential function y is (, ), and its range is (0, ). Because the logarithmic function y log () is the inverse of y, the domain of y log () is (0, ), and its range is (, ). CAUTION Because the domain of y log a () is (0, ) for any a 0 and a, epressions such as log ( 4), log 3 (0), and ln( ) are undefined. Graphing Logarithmic Functions In Chapter we saw that the graphs of a function and its inverse function are symmetric about the line y. Because the logarithm functions are inverses of eponential functions, their graphs are also symmetric about y. E X A M P L E 4 A logarithmic function with base greater than Sketch the graph of g() log () and compare it to the graph of y.

3 . Logarithmic Functions and Their Applications ( 5) 659 calculator close-up The graphs of y ln() and y e are symmetric with respect to the line y. Logarithmic functions with bases other than e and 0 will be graphed on a calculator in Section Make a table of ordered pairs for g() log () using positive numbers for : g() log () 0 3 Draw a curve through these points as shown in Fig..9. The graph of the inverse function y is also shown in Fig..9 for comparison. Note the symmetry of the two curves about the line y. y = y y = y g() = log () (, 0) f() = log a () (a > ) FIGURE.0 FIGURE.9 All logarithmic functions with the base greater than have graphs that are similar to the one in Fig..9. In general, the graph of f() log a () for a has the following characteristics (see Fig..0):. The -intercept of the curve is (, 0).. The domain is (0, ), and the range is (, ). 3. The curve approaches the negative y-ais but does not touch it. 4. The y-values are increasing as we go from left to right along the curve. E X A M P L E 5 A logarithmic function with base less than Sketch the graph of f() log () and compare it to the graph of y. Make a table of ordered pairs for f() log () using positive numbers for : y 4 3 y = y = f() = log / () FIGURE f() log () 0 3 The curve through these points is shown in Fig... The graph of the inverse function y is also shown in Fig.. for comparison. Note the symmetry with respect to the line y. All logarithmic functions with the base between 0 and have graphs that are similar to the one in Fig... In general, the graph of f() log a () for 0 a

4 660 ( 6) Chapter Eponential and Logarithmic Functions y f() = log a () (0 < a < ) (, 0) FIGURE. E X A M P L E 6 E X A M P L E 7 has the following characteristics (see Fig..):. The -intercept of the curve is (, 0).. The domain is (0, ), and the range is (, ). 3. The curve approaches the positive y-ais but does not touch it. 4. The y-values are decreasing as we go from left to right along the curve. Figures.9 and. illustrate the fact that y log a () and y a are inverse functions for any base a. For any given eponential or logarithmic function the inverse function can be easily obtained from the definition of logarithm. Inverses of logarithmic and eponential functions Find the inverse of each function. a) f() 0 b) g() log 3 () a) To find any inverse function we switch the roles of and y. So y 0 becomes 0 y. Now 0 y is equivalent to y log 0 (). So the inverse of f() 0 is y log() or f () log(). b) In g() log 3 ()ory log 3 () we switch and y to get log 3 (y). Now log 3 (y) is equivalent to y 3. So the inverse of g() log 3 () is y 3 or g () 3. Logarithmic Equations In Section. we learned that the eponential functions are one-to-one functions. Because logarithmic functions are inverses of eponential functions, they are oneto-one functions also. For a base-a logarithmic function one-to-one means that if the base-a logarithms of two numbers are equal, then the numbers are equal. One-to-One Property of Logarithms For a 0anda, if log a (m) log a (n), then m n. The one-to-one property of logarithms and the definition of logarithms are the two basic tools that we use to solve equations involving logarithms. We use these tools in the net eample. Logarithmic equations Solve each equation. a) log 3 () b) log (8) 3 c) log( ) log(4) a) Use the definition of logarithms to rewrite the logarithmic equation as an equivalent eponential equation: log 3 () 3 Definition of logarithm 9 Because 3 9 or log 3 9, the solution set is 9.

5 . Logarithmic Functions and Their Applications ( 7) 66 study tip Establish a regular routine of eating, sleeping, and eercise. The ability to concentrate depends on adequate sleep, decent nutrition, and the physical well-being that comes with eercise. b) Use the definition of logarithms to rewrite the logarithmic equation as an equivalent eponential equation: log (8) Definition of logarithm ( 3 ) 8 Raise each side to the power Odd-root property Because or log (8) 3 the solution set is. c) To write an equation equivalent to log( ) log(4), we use the one-to-one property of logarithms: log( ) log(4) 4 One-to-one property of logarithms Even-root property If, then 4 and log(4) log(4). The solution set is {, }. helpful hint The rule of 70 is used to find approimately how long it takes money to double. Divide 70 by the interest rate, ignoring the percent symbol. For eample, at 7% money doubles in approimately 7 0 or 7 0 years. To find the time more eactly, divide ln() by the interest rate. E X A M P L E 8 CAUTION If we have equality of two logarithms with the same base, we use the one-to-one property to eliminate the logarithms. If we have an equation with only one logarithm, such as log a () y, we use the definition of logarithm to write a y and to eliminate the logarithm. Applications When money earns interest compounded continuously, the formula t r ln A P epresses the relationship between the time in years t, the annual interest rate r, the principal P, and the amount A. This formula is used to determine how long it takes for a deposit to grow to a specific amount. Finding the time for a specified growth How long does it take $80 to grow to $40 at % compounded continuously? Use r 0., P $80, and A $40 in the formula, and use a calculator to evaluate the logarithm: t 0. ln l n( 3) It takes approimately 9.55 years, or 9 years and 57 days.

6 66 ( 8) Chapter Eponential and Logarithmic Functions WARM-UPS True or false? Eplain your answer.. The equation a 3 is equivalent to log a () 3.. If (a, b) satisfies y 8, then (a, b) satisfies y log 8 (). 3. If f() a for a 0 and a, then f () log a (). 4. If f() ln(), then f () e. 5. The domain of f() log 6 () is(, ). 6. log 5 (5) 7. log( 0) 8. log(0) log (5) log (3) 5. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is the inverse function for the function f()?. What is log a ()? 3. What is the difference between the common logarithm and the natural logarithm? 4. What is the domain of f() log a ()? 5. What is the one-to-one property of logarithmic functions? 6. What is the relationship between the graphs of f() a and f () log a () for a 0 and a? Write each eponential equation as a logarithmic equation and each logarithmic equation as an eponential equation. See Eample. 7. log (8) 3 8. log 0 (0) y log 5 (). m log b (N) 3. a b 4. a 3 c 5. log 3 () 0 6. log c (t) 4 7. e 3 8. m e Evaluate each logarithm. See Eamples and log (4) 0. log (). log (6). log 4 (6) 3. log (64) 4. log 8 (64) 5. log 4 (64) 6. log 64 (64) 7. log 4 8. log 8 9. log(00) 30. log() 3. log(0.0) 3. log(0,000) 33. log log log 3 (7) 36. log 3 () 37. log 5 (5) 38. log 6 (4) 39. ln(e ) 40. ln e Use a calculator to evaluate each logarithm. Round answers to four decimal places. 4. log(5) 4. log(0.03) 43. ln(6.38) 44. ln(0.3)

7 . Logarithmic Functions and Their Applications ( 9) 663 Sketch the graph of each function. See Eamples 4 and f() log 3 () 46. g() log 0 () Solve each equation. See Eample log() log() log (36) 66. log (00) 67. log (5) 68. log (6) 47. y log 4 () 48. y log 5 () 69. log( ) log(9) 70. ln( 3) ln( ) Use a calculator to solve each equation. Round answers to four decimal places h() log 4 () 50. y log 3 () 75. e e Solve each problem. See Eample 8. Use a calculator as necessary. 77. Double your money. How long does it take $5000 to grow to $0,000 at % compounded continuously? 5. y log 5 () 5. y log 6 () 78. Half the rate. How long does it take $5000 to grow to $0,000 at 6% compounded continuously? 79. Earning interest. How long does it take to earn $000 in interest on a deposit of $6000 at 8% compounded continuously? 80. Lottery winnings. How long does it take to earn $000 interest on a deposit of one million dollars at 9% compounded continuously? Find the inverse of each function. See Eample f() f() f() ln() 56. f() log() 57. f() log () 58. f() log 4 () The annual growth rate for an investment that is growing continuously is given by r t ln A P, where P is the principal and A is the amount after t years. 8. Top stock. An investment of $0,000 in Dell Computer stock in 995 grew to $3,800 in 998. a) Assuming the investment grew continuously, what was the annual growth rate?

8 664 ( 0) Chapter Eponential and Logarithmic Functions 8 b) If Dell continues to grow at the same rate, then what will the $0,000 investment be worth in 00? 8. Chocolate bars. An investment of $0,000 in 980 in Hershey stock was worth $563,000 in 998. Assuming the investment grew continuously, what was the annual growth rate? In chemistry the ph of a solution is defined by ph log 0 [H ], where H is the hydrogen ion concentration of the solution in moles per liter. Distilled water has a ph of approimately 7. A solution with a ph under 7 is called an acid, and one with a ph over 7 is called a base. 83. Tomato juice. Tomato juice has a hydrogen ion concentration of 0 4. mole per liter (mol/l). Find the ph of tomato juice. 84. Stomach acid. The gastric juices in your stomach have a hydrogen ion concentration of 0 mol/l. Find the ph of your gastric juices. 85. Neuse River ph. The ph of a water sample is one of the many measurements of water quality done by the U.S. Geological Survey. The hydrogen ion concentration of the water in the Neuse River at New Bern, North Carolina, was mol/l on July 9, 998 (Water Resources for North Carolina, wwwnc.usgs.gov). What was the ph of the water at that time? 86. Roanoke River ph. On July 9, 998 the hydrogen ion concentration of the water in the Roanoke River at Janesville, North Carolina, was mol/l (Water Resources for North Carolina, wwwnc.usgs.gov). What was the ph of the water at that time? Roanoke River at Janesville where I is the intensity of the sound in watts per square meter. If the intensity of the sound at a rock concert is 0.00 watt per square meter at a distance of 75 meters from the stage, then what is the level of the sound at this point in the audience? 88. Logistic growth. If a rancher has one cow with a contagious disease in a herd of 000, then the time in days t for n of the cows to become infected is modeled by t 5 ln 0 00 n 99 9n. Find the number of days that it takes for the disease to spread to 00, 00, 998, and 999 cows. This model, called a logistic growth model, describes how a disease can spread very rapidly at first and then very slowly as nearly all of the population has become infected. Time (days) t n Number of infected cows FIGURE FOR EXERCISE 88 GETTING MORE INVOLVED 89. Discussion. Use the switch-and-solve method from Chapter to find the inverse of the function f() 5 log ( 3). State the domain and range of the inverse function. 90. Discussion. Find the inverse of the function f() e 4. State the domain and range of the inverse function. ph (standard units) 6 4 GRAPHING CALCULATOR EXERCISES 9. Composition of inverses. Graph the functions y ln(e ) and y e ln(). Eplain the similarities and differences between the graphs July 998 FIGURE FOR EXERCISE 86 Solve each problem. 87. Sound level. The level of sound in decibels (db) is given by the formula L 0 log(i 0 ), 9. The population bomb. The population of the earth is growing continuously with an annual rate of about.6%. If the present population is 6 billion, then the function y 6e 0.06 gives the population in billions years from now. Graph this function for What will the population be in 00 years and in 00 years?