5.4 Logarithmic Functions

Transcription

1 SECTION 5.4 Logarithmic Functions 8 Problems and provide definitions for two other transcendental functions.. The hperbolic sine function, designated b sinh, is defined as sinh = e - e - (a) Show that f = sinh is an odd function. (b) Graph f = sinh using a graphing utilit.. The hperbolic cosine function, designated b cosh, is defined as cosh = e + e - (b) Graph f = cosh using a graphing utilit. (c) Refer to Problem. Show that, for ever, cosh - sinh = 4. Historical Problem Pierre de Fermat (60 665) conjectured that the function f = + for =,,, Á, would alwas have a value equal to a prime number. But Leonhard Euler (707 78) showed that this formula fails for = 5. Use a calculator to determine the prime numbers produced b f for =,,, 4. Then show that f5 = 64 * 6,700,47, which is not prime. (a) Show that f = cosh is an even function. Eplaining Concepts: Discussion and Writing 5. The bacteria in a 4-liter container double ever minute. After 60 minutes the container is full. How long did it take to fill half the container? 6. Eplain in our own words what the number e is. Provide at least two applications that use this number. 7. Do ou think that there is a power function that increases more rapidl than an eponential function whose base is greater than? Eplain. 8. As the base a of an eponential function f = a, where a 7 increases, what happens to the behavior of its graph for 7 0? What happens to the behavior of its graph for 6 0? 9. The graphs of and = a = a - are identical. Wh? a b Are You Prepared? Answers. 64; 4;. {-4, }. False True Logarithmic Functions PREPARING FOR THIS SECTION Before getting started, review the following: Solving Inequalities (Appendi A, Section A.9, pp. A75 A78) Quadratic Inequalities (Section.5, pp ) Now Work the Are You Prepared? problems on page 9. Polnomial and Rational Inequalities (Section 4.4, pp. 4 7) Solving Equations (Appendi A, Section A.6, pp. A44 A46) OBJECTIVES Change Eponential Statements to Logarithmic Statements and Logarithmic Statements to Eponential Statements (p. 84) Evaluate Logarithmic Epressions (p. 84) Determine the Domain of a Logarithmic Function (p. 85) 4 Graph Logarithmic Functions (p. 86) 5 Solve Logarithmic Equations (p. 90) Recall that a one-to-one function = f has an inverse function that is defined (implicitl) b the equation = f. In particular, the eponential function = f = a, where a 7 0 and a Z, is one-to-one and hence has an inverse function that is defined implicitl b the equation = a, a 7 0, a Z This inverse function is so important that it is given a name, the logarithmic function.

2 84 CHAPTER 5 Eponential and Logarithmic Functions DEFINITION The logarithmic function to the base a, where a 7 0 and a Z, is denoted b = log a (read as is the logarithm to the base a of ) and is defined b In Words When ou read log a, think to ourself a raised to what power gives me. = log a if and onl if = a The domain of the logarithmic function = log a is 7 0. As this definition illustrates, a logarithm is a name for a certain eponent. So, log a represents the eponent to which a must be raised to obtain. EXAMPLE Relating Logarithms to Eponents (a) If = log then =,. For eample, the logarithmic statement 4 = log 8 is equivalent to the eponential statement 8 = 4. (b) If = log then For eample, - = log is equivalent to 5 a 5, = 5. 5 = b Change Eponential Statements to Logarithmic Statements and Logarithmic Statements to Eponential Statements We can use the definition of a logarithm to convert from eponential form to logarithmic form, and vice versa, as the following two eamples illustrate. EXAMPLE Changing Eponential Statements to Logarithmic Statements Change each eponential statement to an equivalent statement involving a logarithm. (a). = m (b) e b = 9 (c) a 4 = 4 Use the fact that = log and = a a, where a 7 0 and a Z, are equivalent. (a) If. = m, then = log. m. (b) If e b = 9, then b = log e 9. (c) If a 4 = 4, then 4 = log a 4. Now Work PROBLEM 9 EXAMPLE Changing Logarithmic Statements to Eponential Statements Change each logarithmic statement to an equivalent statement involving an eponent. (a) log a 4 = 5 (b) log e b = - (c) log 5 = c (a) If log then a 5 a 4 = 5, = 4. (b) If log then e - e b = -, = b. (c) If log then c 5 = c, = 5. Now Work PROBLEM 7 Evaluate Logarithmic Epressions To find the eact value of a logarithm, we write the logarithm in eponential notation using the fact that = log is equivalent to a a = and use the fact that if a u = a v, then u = v.

3 SECTION 5.4 Logarithmic Functions 85 EXAMPLE 4 Finding the Eact Value of a Logarithmic Epression Find the eact value of: (a) log 6 (b) log 7 (a) To evaluate log 6, think raised to what power ields 6. So, = log 6 = 6 = 4 = 4 Change to eponential form. 6 = 4 Equate eponents. Therefore, log 6 = 4. Now Work PROBLEM 5 (b) To evaluate log, think raised 7 to what power ields. So, 7 = log 7 = Change to eponential 7 form. = - 7 = = - = - Equate eponents. Therefore, log 7 = -. Determine the Domain of a Logarithmic Function The logarithmic function = log a has been defined as the inverse of the eponential function = a. That is, if f = a, then f - = log a. Based on the discussion given in Section 5. on inverse functions, for a function f and its inverse f -, we have Domain of f - = Range of f and Range of f - = Domain of f Consequentl, it follows that Domain of the logarithmic function = Range of the eponential function = 0, q Range of the logarithmic function = Domain of the eponential function = - q, q In the net bo, we summarize some properties of the logarithmic function: = log a defining equation: = a Domain: q Range: - q 6 6 q The domain of a logarithmic function consists of the positive real numbers, so the argument of a logarithmic function must be greater than zero. EXAMPLE 5 Finding the Domain of a Logarithmic Function Find the domain of each logarithmic function. (a) F = log + (b) g = log 5 a + (c) h = log > ƒƒ - b (a) The domain of F consists of all for which + 7 0, that is, 7 -. Using interval notation, the domain of f is -, q. (b) The domain of g is restricted to Solving this inequalit, we find that the domain of g consists of all between - and, that is, or, using interval notation, -,.

4 86 CHAPTER 5 Eponential and Logarithmic Functions (c) Since ƒƒ 7 0, provided that Z 0, the domain of h consists of all real numbers ecept zero or, using interval notation, - q, 0 0, q. Now Work PROBLEMS 9 AND 45 4 Graph Logarithmic Functions Since eponential functions and logarithmic functions are inverses of each other, the graph of the logarithmic function = log a is the reflection about the line = of the graph of the eponential function = a, as shown in Figure 0. Figure 0 a (, a ) (a, ) (0, ) (, a) (0, ) (, a ) a (, a) log a (a, ) (, 0) (a) 0 a ( a, ) log a (, 0) (b) a ( a,) For eample, to graph = log graph =, and reflect it about the line See Figure. To graph graph = a =. = log >, and reflect it about the b line =. See Figure. Figure (, ) (0, ) (, ) log (, ) Figure ( ) (, ) (0, ) (, ) (, ) (, 0) (, ) (, 0) (, ) log / Now Work PROBLEM 59 The graphs of = log a in Figures 0(a) and (b) lead to the following properties. Properties of the Logarithmic Function f() log a. The domain is the set of positive real numbers or 0, q using interval notation; the range is the set of all real numbers or - q, q using interval notation.. The -intercept of the graph is. There is no -intercept.. The -ais = 0 is a vertical asmptote of the graph. 4. A logarithmic function is decreasing if 0 6 a 6 and increasing if a The graph of f contains the points, 0, a,, and a a, -b. 6. The graph is smooth and continuous, with no corners or gaps.

5 SECTION 5.4 Logarithmic Functions 87 In Words = log e is written = ln If the base of a logarithmic function is the number e, then we have the natural logarithm function. This function occurs so frequentl in applications that it is given a special smbol, ln (from the Latin, logarithmus naturalis). That is, = ln if and onl if = e () Since = ln and the eponential function = e are inverse functions, we can obtain the graph of = ln b reflecting the graph of = e about the line =. See Figure. Using a calculator with an ln ke, we can obtain other points on the graph of f = ln. See Table 7. Seeing the Concept Graph Y = e and Y = ln on the same square screen. Use evalueate to verif the points on the graph given in Figure. Do ou see the smmetr of the two graphs with respect to the line =? Figure Table 7 ln ( (, ) e 0, e 0 (, 0) ( e,) 5 ) 0 (, e) In (e, ) Figure 4 EXAMPLE 6 0 Graphing a Logarithmic Function and Its Inverse (a) Find the domain of the logarithmic function f = -ln -. (b) Graph f. (c) From the graph, determine the range and vertical asmptote of f. (d) Find f -, the inverse of f. (e) Find the domain and the range of f -. (f) Graph f -. (a) The domain of f consists of all for which or, equivalentl, 7. The domain of f is {ƒ 7 } or, q in interval notation. (b) To obtain the graph of = -ln -, we begin with the graph of = ln and use transformations. See Figure 4. 0 (e, ) ( (, 0 ) e, ) ( e, ) ( ) e, (, 0 ) 5 (e, ) (e, ) (, 0 ) (a) In Multipl b ; reflect about -ais (b) In Replace b ; shift right units. (c) In ( ) (c) The range of f = -ln - is the set of all real numbers. The vertical asmptote is =. [Do ou see wh? The original asmptote = 0 is shifted to the right units.]

6 88 CHAPTER 5 Eponential and Logarithmic Functions (d) To find f -, begin with = -ln -.The inverse function is defined (implicitl) b the equation Proceed to solve for. = -ln - - = ln - e - = - = e - + Isolate the logarithm. Change to an eponential statement. Solve for. The inverse of f is f - = e - +. (e) The domain of f - equals the range of f,which is the set of all real numbers,from part (c). The range of f - is the domain of f,which is, q in interval notation. (f) To graph f -, use the graph of f in Figure 4(c) and reflect it about the line =. See Figure 5. We could also graph f - = e - + using transformations. Figure 5 (, e ) 5 f () e (0, ) (, e ) ( e, ) (, 0) f () ln( ) 5 (e, ) Figure Now Work PROBLEM 7 (, ) 0 (0, ) log (, 0) 4 (, 0 ) If the base of a logarithmic function is the number 0, then we have the common logarithm function. If the base a of the logarithmic function is not indicated, it is understood to be 0. That is, = log if and onl if = 0 Since = log and the eponential function = 0 are inverse functions, we can obtain the graph of = log b reflecting the graph of = 0 about the line =. See Figure 6. EXAMPLE 7 Graphing a Logarithmic Function and Its Inverse (a) Find the domain of the logarithmic function f = log -. (b) Graph f. (c) From the graph, determine the range and vertical asmptote of f. (d) Find f -, the inverse of f. (e) Find the domain and the range of f -. (f) Graph f -. (a) The domain of f consists of all for which or, equivalentl, 7. The domain of f is { 7 } or, q in interval notation. (b) To obtain the graph of = log -, begin with the graph of = log and use transformations. See Figure 7.

7 SECTION 5.4 Logarithmic Functions 89 Figure 7 0 (, ) (0, ) (, 0) (, 0) (, ) (, 0) ( 0,) (,) ( 0,) Replace b ; horizontal shift right unit Multipl b ; vertical stretch b a factor of. (a) log (b) log ( ) (c) log ( ) (c) The range of f = log - is the set of all real numbers. The vertical asmptote is =. (d) Begin with = log -. The inverse function is defined (implicitl) b the equation = log - Proceed to solve for. = log - 0 / = - = 0 / + Isolate the logarithm. Change to an eponential statement. Solve for. The inverse of f is f - = 0 / +. (e) The domain of f - is the range of f, which is the set of all real numbers, from part (c).the range of f - is the domain of f, which is, q in interval notation. (f) To graph f -, we use the graph of f in Figure 7(c) and reflect it about the line =. See Figure 8. We could also graph f - = 0 > + using transformations. Figure 8 0 (, ) f () 0 / (, ) (0, ) f () log ( ) (, 0) Now Work PROBLEM 79

8 90 CHAPTER 5 Eponential and Logarithmic Functions 5 Solve Logarithmic Equations Equations that contain logarithms are called logarithmic equations. Care must be taken when solving logarithmic equations algebraicall. In the epression log a M, remember that a and M are positive and a Z. Be sure to check each apparent solution in the original equation and discard an that are etraneous. Some logarithmic equations can be solved b changing the logarithmic equation to eponential form using the fact that = log means a a =. EXAMPLE 8 Solving Logarithmic Equations Solve: (a) log 4-7 = (b) log 64 = (a) We can obtain an eact solution b changing the logarithmic equation to eponential form. log 4-7 = 4-7 = 4-7 = 9 4 = 6 = 4 Change to eponential form using = log a means a =. Check: log 4-7 = log 4 # 4-7 = log 9 = The solution set is {4}. (b) We can obtain an eact solution b changing the logarithmic equation to eponential form. log 64 = = 64 = ;64 = ;8 Change to eponential form. Square Root Method = 9 The base of a logarithm is alwas positive. As a result, we discard -8. We check the solution 8. Check: log 8 64 = The solution set is {8}. 8 = 64 EXAMPLE 9 Using Logarithms to Solve an Eponential Equation Solve: e = 5 We can obtain an eact solution b changing the eponential equation to logarithmic form. The solution set is e = 5 ln 5 = = ln 5 L e ln 5 f. Change to logarithmic form using the fact that if e = then = ln. Eact solution Approimate solution Now Work PROBLEMS 87 AND 99

9 SECTION 5.4 Logarithmic Functions 9 EXAMPLE 0 COMMENT A BAC of 0.0% results in a loss of consciousness in most people. COMMENT Most states use 0.08% or 0.0% as the blood alcohol content at which a DUI citation is given. Alcohol and Driving Blood alcohol concentration (BAC) is a measure of the amount of alcohol in a person s bloodstream. A BAC of 0.04% means that a person has 4 parts alcohol per 0,000 parts blood in the bod. Relative risk is defined as the likelihood of one event occurring divided b the likelihood of a second event occurring. For eample, if an individual with a BAC of 0.0% is.4 times as likel to have a car accident as an individual that has not been drinking, the relative risk of an accident with a BAC of 0.0% is.4. Recent medical research suggests that the relative risk R of having an accident while driving a car can be modeled b an equation of the form R = e k where is the percent of concentration of alcohol in the bloodstream and k is a constant. (a) Research indicates that the relative risk of a person having an accident with a BAC of 0.0% is.4. Find the constant k in the equation. (b) Using this value of k, what is the relative risk if the concentration is 0.7%? (c) Using this same value of k, what BAC corresponds to a relative risk of 00? (d) If the law asserts that anone with a relative risk of 5 or more should not have driving privileges, at what concentration of alcohol in the bloodstream should a driver be arrested and charged with a DUI (driving under the influence)? (a) For a concentration of alcohol in the blood of 0.0% and a relative risk of.4, we let = 0.0 and R =.4 in the equation and solve for k. R = e k.4 = e k k = ln.4 k = Change to a logarithmic statement. Solve for k. (b) For a concentration of 0.7%, we have = 0.7. Using k = 6.8 in the equation, we find the relative risk R to be R = e k = e L 7.5 For a concentration of alcohol in the blood of 0.7%, the relative risk of an accident is about 7.5. That is, a person with a BAC of 0.7% is 7.5 times as likel to have a car accident as a person with no alcohol in the bloodstream. (c) For a relative risk of 00, we have R = 00. Using k = 6.8 in the equation R = e k, we find the concentration of alcohol in the blood obes 00 = e = ln 00 = ln L 6.8 ln L 0.7 R =.4; = 0.0 R = e k, R = 00; k = 6.8 Change to a logarithmic statement. Solve for. For a concentration of alcohol in the blood of 0.7%, the relative risk of an accident is 00. (d) For a relative risk of 5, we have R = 5. Using k = 6.8 in the equation R = e k, we find the concentration of alcohol in the bloodstream obes 5 = e = ln 5 = ln L A driver with a BAC of 0.096% or more should be arrested and charged with DUI.

10 9 CHAPTER 5 Eponential and Logarithmic Functions SUMMARY Properties of the Logarithmic Function f = log a, a 7 Domain: the interval 0, q; Range: the interval - q, q = log means = a a -intercept: ; -intercept: none; vertical asmptote: = 0 (-ais); increasing; one-to-one See Figure 9(a) for a tpical graph. f = log a, 0 6 a 6 Domain: the interval 0, q; Range: the interval - q, q ( = log means = a a ) -intercept: ; -intercept: none; vertical asmptote: = 0 (-ais); decreasing; one-to-one See Figure 9(b) for a tpical graph. Figure 9 0 log a (a, ) (a, ) (, 0) (a) a ( a, ) 0 (, 0) ( a, (b) 0 a ) log a 5.4 Assess Your Understanding Are You Prepared? Answers are given at the end of these eercises. If ou get a wrong answer, read the pages listed in red.. Solve each inequalit: (a) (pp. A75 A78) (b) (pp ) Concepts and Vocabular 4. The domain of the logarithmic function f = log a is. 5. The graph of ever logarithmic function f = log a, where a 7 0 and a Z, passes through three points:,, and. Skill Building -. Solve the inequalit: (pp. 4 7) Solve: + = 9 (pp. A44 A5) 6. If the graph of a logarithmic function f = log a, where a 7 0 and a Z, is increasing, then its base must be larger than. 7. True or False If = log then = a a,. 8. True or False The graph of f = log a, where a 7 0 and a Z, has an -intercept equal to and no -intercept. In Problems 9 6, change each eponential statement to an equivalent statement involving a logarithm = 0. 6 = 4. a =.6. a =.. = = e = 8 6. e. = M In Problems 7 4, change each logarithmic statement to an equivalent statement involving an eponent. 7. log 8 = 8. log a 9 b = - 9. log a = 6 0. log b 4 =. log =. log 6 =. ln 4 = 4. ln = 4

11 SECTION 5.4 Logarithmic Functions 9 In Problems 5 6, find the eact value of each logarithm without using a calculator. 5. log 6. log log log a 9 b 9. log > 6 0. log > 9. log 0 0. log 5 5. log 4 4. log 9 5. lne 6. ln e In Problems 7 48, find the domain of each function. 7. f = ln - 8. g = ln - 9. F = log 40. H = log 5 4. f = - log 4 c - 5 d 4. g = ln + 4. f = lna + b 44. g = lna - 5 b 45. g = log 5a + b 46. h = log a b 47. f = ln 48. g = - ln In Problems 49 56, use a calculator to evaluate each epression. Round our answer to three decimal places. 49. ln ln 5 5. ln ln ln 4 + ln log 4 + log 54. log 5 + log 0 ln 5 + ln ln 5 + log 50 log 4 - ln 56. log 80 - ln 5 log 5 + ln Find a so that the graph of f = log a contains the point,. 58. Find a so that the graph of f = log contains the point a a, -4b. In Problems 59 6, graph each function and its inverse on the same Cartesian plane. 59. f = ; f - = log 60. f = 4 ; f - = log 4 6. f = a b ; f - = log 6. f = a ; b f - = log In Problems 6 70, the graph of a logarithmic function is given. Match each graph to one of the following functions: (a) = log (b) = log - (c) = -log (d) = -log - (e) = log - (f) = log - (g) = log - (h) = - log In Problems 7 86, use the given function f to: (a) Find the domain of f. (b) Graph f. (c) From the graph, determine the range and an asmptotes of f. (d) Find f -, the inverse of f. (e) Find the domain and the range of f -. (f) Graph f f = ln f = ln - 7. f = + ln 74. f = -ln-

12 94 CHAPTER 5 Eponential and Logarithmic Functions 75. f = ln f = - ln f = log f = log f = log 80. f = log- 8. f = + log + 8. f = - log + 8. f = e f = e f = / f = - + In Problems 87 0, solve each equation. 87. log = 88. log 5 = 89. log + = 90. log - = 9. log 4 = 9. log a 8 b = 9. ln e = ln e - = log 4 64 = 96. log 5 65 = 97. log 4 = log 6 6 = e = e - = 0. e + 5 = 8 0. e - + = 0. log + = 04. log = 05. log 8 = log = e 0. = # 0-7 = 09. # 0 - = e + = 5 Mied Practice. Suppose that G = log + -. (a) What is the domain of G? (b) What is G(40)? What point is on the graph of G? (c) If G =, what is? What point is on the graph of G? (d) What is the zero of G?. Suppose that F = log + -. (a) What is the domain of F? (b) What is F(7)? What point is on the graph of F? (c) If F = -, what is? What point is on the graph of F? (d) What is the zero of F? In Problems 6, graph each function. Based on the graph, state the domain and the range and find an intercepts. ln- if 6 0. f = e ln if 7 0 -ln if f = e ln if Ú ln- if - 4. f = e -ln- if ln if f = e -ln if Ú Applications and Etensions 7. Chemistr The ph of a chemical solution is given b the formula ph = -log 0 [H + ] where [H + ] is the concentration of hdrogen ions in moles per liter. Values of ph range from 0 (acidic) to 4 (alkaline). (a) What is the ph of a solution for which [H + ] is 0.? (b) What is the ph of a solution for which [H + ] is 0.0? (c) What is the ph of a solution for which [H + ] is 0.00? (d) What happens to ph as the hdrogen ion concentration decreases? (e) Determine the hdrogen ion concentration of an orange (ph =.5). (f) Determine the hdrogen ion concentration of human blood (ph = 7.4). 8. Diversit Inde Shannon s diversit inde is a measure of the diversit of a population. The diversit inde is given b the formula H = -p log p + p log p + Á + p n log p n where p is the proportion of the population that is species, p is the proportion of the population that is species, and so on. (a) According to the U.S. Census Bureau, the distribution of race in the United States in 007 was as follows: Race Proportion American Indian or Native Alaskan 0.05 Asian 0.04 Black or African American 0.9 Hispanic 0.5 Native Hawaiian or Pacific Islander 0.00 White Source: U.S. Census Bureau Compute the diversit inde of the United States in 007.

13 SECTION 5.4 Logarithmic Functions 95 (b) The largest value of the diversit inde is given b H ma = logs, where S is the number of categories of race. Compute H ma. (c) The evenness ratio is given b E H = H, where H ma 0 E H. If E H =, there is complete evenness. Compute the evenness ratio for the United States. (d) Obtain the distribution of race for the United States in 00 from the Census Bureau. Compute Shannon s diversit inde. Is the United States becoming more diverse? Wh? 9. Atmospheric Pressure The atmospheric pressure p on an object decreases with increasing height. This pressure, measured in millimeters of mercur, is related to the height h (in kilometers) above sea level b the function p(h) = 760e -0.45h (a) Find the height of an aircraft if the atmospheric pressure is 0 millimeters of mercur. (b) Find the height of a mountain if the atmospheric pressure is 667 millimeters of mercur. 0. Healing of Wounds The normal healing of wounds can be modeled b an eponential function. If A 0 represents the original area of the wound and if A equals the area of the wound, then the function A(n) = A 0 e -0.5n describes the area of a wound after n das following an injur when no infection is present to retard the healing. Suppose that a wound initiall had an area of 00 square millimeters. (a) If healing is taking place, after how man das will the wound be one-half its original size? (b) How long before the wound is 0% of its original size?. Eponential Probabilit Between :00 PM and :00 PM, cars arrive at Citibank s drive-thru at the rate of 6 cars per hour (0. car per minute). The following formula from statistics can be used to determine the probabilit that a car will arrive within t minutes of :00 PM. Ft = - e -0.t (a) Determine how man minutes are needed for the probabilit to reach 50%. (b) Determine how man minutes are needed for the probabilit to reach 80%. (c) Is it possible for the probabilit to equal 00%? Eplain.. Eponential Probabilit Between 5:00 PM and 6:00 PM, cars arrive at Jiff Lube at the rate of 9 cars per hour (0.5 car per minute). The following formula from statistics can be used to determine the probabilit that a car will arrive within t minutes of 5:00 PM. Ft = - e -0.5t (a) Determine how man minutes are needed for the probabilit to reach 50%. (b) Determine how man minutes are needed for the probabilit to reach 80%.. Drug Medication The formula D = 5e -0.4h can be used to find the number of milligrams D of a certain drug that is in a patient s bloodstream h hours after the drug was administered. When the number of milligrams reaches, the drug is to be administered again. What is the time between injections? 4. Spreading of Rumors A model for the number N of people in a college communit who have heard a certain rumor is N = P - e -0.5d where P is the total population of the communit and d is the number of das that have elapsed since the rumor began. In a communit of 000 students, how man das will elapse before 450 students have heard the rumor? 5. Current in a RL Circuit The equation governing the amount of current I (in amperes) after time t (in seconds) in a simple RL circuit consisting of a resistance R (in ohms), an inductance L (in henrs), and an electromotive force E (in volts) is I = E R - e-r>lt 4 If E = volts, R = 0 ohms, and L = 5 henrs, how long does it take to obtain a current of 0.5 ampere? Of.0 ampere? Graph the equation. 6. Learning Curve Pschologists sometimes use the function Lt = A - e -kt to measure the amount L learned at time t. The number A represents the amount to be learned, and the number k measures the rate of learning. Suppose that a student has an amount A of 00 vocabular words to learn. A pschologist determines that the student learned 0 vocabular words after 5 minutes. (a) Determine the rate of learning k. (b) Approimatel how man words will the student have learned after 0 minutes? (c) After 5 minutes? (d) How long does it take for the student to learn 80 words? Loudness of Sound Problems 7 0 use the following discussion: The loudness L, measured in decibels (db), of a sound of intensit, measured in watts per square meter, is defined as L = 0 log, where I 0 = 0 - watt per square meter is the least intense I 0 sound that a human ear can detect. Determine the loudness, in decibels, of each of the following sounds. 7. Normal conversation: intensit of = 0-7 watt per square 9. Heav cit traffic: intensit of = 0 - watt per square meter. meter. 8. Amplified rock music: intensit of 0 - watt per square meter. 0. Diesel truck traveling 40 miles per hour 50 feet awa: intensit 0 times that of a passenger car traveling 50 miles per hour 50 feet awa whose loudness is 70 decibels.

14 96 CHAPTER 5 Eponential and Logarithmic Functions The Richter Scale Problems and use the following discussion: The Richter scale is one wa of converting seismographic readings into numbers that provide an eas reference for measuring the magnitude M of an earthquake. All earthquakes are compared to a zero-level earthquake whose seismographic reading measures 0.00 millimeter at a distance of 00 kilometers from the epicenter. An earthquake whose seismographic reading measures millimeters has magnitude M, given b M = log 0 where 0 = 0 - is the reading of a zero-level earthquake the same distance from its epicenter. In Problems and, determine the magnitude of each earthquake.. Magnitude of an Earthquake Meico Cit in 985: seismographic reading of 5,89 millimeters 00 kilometers from the center. Magnitude of an Earthquake San Francisco in 906: seismographic reading of 50,9 millimeters 00 kilometers from the center. Alcohol and Driving The concentration of alcohol in a person s bloodstream is measurable. Suppose that the relative risk R of having an accident while driving a car can be modeled b an equation of the form R = e k where is the percent of concentration of alcohol in the bloodstream and k is a constant. Eplaining Concepts: Discussion and Writing (a) Suppose that a concentration of alcohol in the bloodstream of 0.0 percent results in a relative risk of an accident of.4. Find the constant k in the equation. (b) Using this value of k, what is the relative risk if the concentration is 0.7 percent? (c) Using the same value of k, what concentration of alcohol corresponds to a relative risk of 00? (d) If the law asserts that anone with a relative risk of having an accident of 5 or more should not have driving privileges, at what concentration of alcohol in the bloodstream should a driver be arrested and charged with a DUI? (e) Compare this situation with that of Eample 0. If ou were a lawmaker, which situation would ou support? Give our reasons. 4. Is there an function of the form = a, 0 6 a 6, that increases more slowl than a logarithmic function whose base is greater than? Eplain. 5. In the definition of the logarithmic function, the base a is not allowed to equal. Wh? 6. Critical Thinking In buing a new car, one consideration might be how well the price of the car holds up over time. Different makes of cars have different depreciation rates. One wa to compute a depreciation rate for a car is given here. Suppose that the current prices of a certain automobile are as shown in the table. Age in Years New 4 5 $8,000 $6,600 $,400 $8,750 $5,400 $,00 Use the formula New = Olde Rt to find R, the annual depreciation rate, for a specific time t. When might be the best time to trade in the car? Consult the NADA ( blue ) book and compare two like models that ou are interested in. Which has the better depreciation rate? Are You Prepared? Answers. (a) (b) 6 - or or 7. {} 5.5 Properties of Logarithms OBJECTIVES Work with the Properties of Logarithms (p. 96) Write a Logarithmic Epression as a Sum or Difference of Logarithms (p. 98) Write a Logarithmic Epression as a Single Logarithm (p. 99) 4 Evaluate Logarithms Whose Base Is Neither 0 Nor e (p. 0) Work with the Properties of Logarithms Logarithms have some ver useful properties that can be derived directl from the definition and the laws of eponents.