3.3 Logarithmic Functions and Their Graphs
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1 274 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions What ou ll learn about Inverses of Eponential Functions Common Logarithms Base 0 Natural Logarithms Base e Graphs of Logarithmic Functions Measuring Sound Using Decibels... and wh Logarithmic functions are used in man applications, including the measurement of the relative intensit of sounds. 3.3 Logarithmic Functions and Their Graphs Inverses of Eponential Functions In Section.4 we learned that, if a function passes the horizontal line test, then the inverse of the function is also a function. Figure 3.8 shows that an eponential function ƒ2 = b would pass the horizontal line test. So it has an inverse that is a function. This inverse is the logarithmic function with base b, denoted log b 2, or more simpl as log b. That is, if ƒ2 = b with b 7 0 and b Z, then ƒ - 2 = log b. See Figure 3.9. = b b > = b 0 < b < (a) (b) FIGURE 3.8 Eponential functions are either (a) increasing or (b) decreasing. = b = An immediate and useful consequence of this definition is the link between an eponential equation and its logarithmic counterpart. = log b Changing Between Logarithmic and Eponential Form If 7 0 and 0 6 b Z, then = log if and onl if b b 2 =. FIGURE 3.9 Because logarithmic functions are inverses of eponential functions, we can obtain the graph of a logarithmic function b the mirror or rotational methods discussed in Section.4. A Bit of Histor Logarithmic functions were developed around 594 as computational tools b Scottish mathematician John Napier (550 67). He originall called them artificial numbers, but changed the name to logarithms, which means reckoning numbers. Generall b> In practice, logarithmic bases are almost alwas greater than. This linking statement sas that a logarithm is an eponent. Because logarithms are eponents, we can evaluate simple logarithmic epressions using our understanding of eponents. EXAMPLE Evaluating Logarithms (a) log because = 3 = 8. (b) log because 3 /2 3 3 = /2 = 3. (c) log because 5-2 =. 5 2 = 5 25 = (d) log because = 0 =. (e) log because = = 7. Now tr Eercise. We can generalize the relationships observed in Eample. Basic Properties of Logarithms For 0 6 b Z, 7 0, and an real number, log because b 0 b = 0 =. log because b b b = = b. log because b = b b b =. b log b = because log b = log b.
2 SECTION 3.3 Logarithmic Functions and Their Graphs 275 These properties give us efficient was to evaluate simple logarithms and some eponential epressions. The first two parts of Eample 2 are the same as the first two parts of Eample. EXAMPLE 2 Evaluating Logarithmic and Eponential Epressions (a) log 2 8 = log = 3. (b) log 3 3 = log 3 3 /2 = /2. (c) 6 log6 =. Now tr Eercise 5. Logarithmic functions are inverses of eponential functions. So the inputs and outputs are switched. Table 3.6 illustrates this relationship for ƒ2 = 2 and f - 2 = log 2. Table 3.6 An Eponential Function and Its Inverse ƒ2 = 2 f - 2 = log 2-3 /8 / /4 /4-2 - /2 / This relationship can be used to produce both tables and graphs for logarithmic functions, as ou will discover in Eploration. EXPLORATION Comparing Eponential and Logarithmic Functions. Set our grapher to Parametric mode and Simultaneous graphing mode. Set XT = T and YT = 2^T. Set X2T = 2^T and Y2T = T. Creating Tables. Set TblStart = -3 and Tbl =. Use the Table feature of our grapher to obtain the decimal form of both parts of Table 3.6. Be sure to scroll to the right to see X2T and Y2T. Drawing Graphs. Set Tmin = -6, Tma = 6, and Tstep = 0.5. Set the, 2 window to 3-6, 64 b 3-4, 44. Use the Graph feature to obtain the simultaneous graphs of ƒ2 = 2 and f - 2 = log 2. Use the Trace feature to eplore the numerical relationships within the graphs. 2. Graphing in Function mode. Graph = 2 in the same window. Then use the draw inverse command to draw the graph of = log 2. Common Logarithms Base 0 Logarithms with base 0 are called common logarithms. Because of their connection to our base-ten number sstem, the metric sstem, and scientific notation, common logarithms are especiall useful. We often drop the subscript of 0 for the base when using common logarithms. The common logarithmic function log 0 = log is the inverse of the eponential function ƒ2 = 0. So = log if and onl if 0 =. Appling this relationship, we can obtain other relationships for logarithms with base 0.
3 276 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions Basic Properties of Common Logarithms Let and be real numbers with 7 0. log = 0 because 0 0 =. log 0 = because 0 = 0. log 0 = because 0 = 0. 0 log = because log = log. Some Words of Warning In Figure 3.20, notice we used 0^Ans instead of 0^ to check log This is because graphers generall store more digits than the displa and so we can obtain a more accurate check. Even so, because log is an irrational number, a grapher cannot produce its eact value, so checks like those shown in Figure 3.20 ma not alwas work out so perfectl. log(34.5) 0^Ans log(0.43) 0^Ans FIGURE 3.20 Doing and checking common logarithmic computations. (Eample 4) Using the definition of common logarithm or these basic properties, we can evaluate epressions involving a base of 0. EXAMPLE 3 Evaluating Logarithmic and Eponential Epressions Base 0 (a) log 00 = log because = 2 = 00. (b) log 25 0 = log 0 /5 =. 5 (c) log. 000 = log 0 3 = log 0-3 = -3 (d) 0 log 6 = 6. Now tr Eercise 7. ke on a calculator, as illus- Common logarithms can be evaluated b using the trated in Eample 4. LOG EXAMPLE 4 Evaluating Common Logarithms with a Calculator Use a calculator to evaluate the logarithmic epression if it is defined, and check our result b evaluating the corresponding eponential epression. (a) log 34.5 =.537 Á because 0.537Á = (b) log 0.43 = Á because Á = See Figure (c) log -32 is undefined because there is no real number such that 0 = -3. A grapher will ield either an error message or a comple-number answer for entries such as log -32. We shall restrict the domain of logarithmic functions to the set of positive real numbers and ignore such comple-number answers. Now tr Eercise 25. Changing from logarithmic form to eponential form sometimes is enough to solve an equation involving logarithmic functions. EXAMPLE 5 Solving Simple Logarithmic Equations Solve each equation b changing it to eponential form. (a) log = 3 (b) log 2 = 5 SOLUTION (a) Changing to eponential form, = 0 3 = 000. (b) Changing to eponential form, = 2 5 = 32. Now tr Eercise 33.
4 SECTION 3.3 Logarithmic Functions and Their Graphs 277 Reading a Natural Log The epression ln is pronounced el en of e. The l is for logarithm, and the n is for natural. Natural Logarithms Base e Because of their special calculus properties, logarithms with the natural base e are used in man situations. Logarithms with base e are natural logarithms. We often use the special abbreviation ln (without a subscript) to denote a natural logarithm. Thus, the natural logarithmic function log e = ln. It is the inverse of the eponential function ƒ2 = e. So = ln if and onl if e =. Appling this relationship, we can obtain other fundamental relationships for logarithms with the natural base e. Basic Properties of Natural Logarithms Let and be real numbers with 7 0. ln = 0 because e 0 =. ln e = because e = e. ln e = because e = e. e ln = because ln = ln. Using the definition of natural logarithm or these basic properties, we can evaluate epressions involving the natural base e. EXAMPLE 6 Evaluating Logarithmic and Eponential Epressions Base e (a) ln 2e = log because e /2 e 2e = /2 = 2e. (b) ln e 5 = log e e 5 = 5. (c) e ln 4 = 4. Now tr Eercise 3. ke on a calculator, as illus- Natural logarithms can be evaluated b using the trated in Eample 7. LN ln(23.5) e^ans ln(0.48) e^ans FIGURE 3.2 Doing and checking natural logarithmic computations. (Eample 7) EXAMPLE 7 Evaluating Natural Logarithms with a Calculator Use a calculator to evaluate the logarithmic epression, if it is defined, and check our result b evaluating the corresponding eponential epression. (a) ln 23.5 = 3.57 Á because e 3.57Á = (b) ln 0.48 = Á because e Á = See Figure 3.2. (c) ln -52 is undefined because there is no real number such that e = -5. A grapher will ield either an error message or a comple-number answer for entries such as ln -52. We will continue to restrict the domain of logarithmic functions to the set of positive real numbers and ignore such comple-number answers. Now tr Eercise 29. Graphs of Logarithmic Functions The natural logarithmic function ƒ2 = ln is one of the basic functions introduced in Section.3. We now list its properties.
5 278 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions BASIC FUNCTION The Natural Logarithmic Function [ 2, 6] b [ 3, 3] FIGURE 3.22 ƒ2 = ln Domain: 0, q2 Range: All reals Continuous on 0, q2 Increasing on 0, q2 No smmetr Not bounded above or below No local etrema No horizontal asmptotes Vertical asmptote: = 0 End behavior: lim ln = q : q An logarithmic function g2 = log b with b 7 has the same domain, range, continuit, increasing behavior, lack of smmetr, and other general behavior as ƒ2 = ln. It is rare that we are interested in logarithmic functions g2 = log b with 0 6 b 6. So, the graph and behavior of ƒ2 = ln are tpical of logarithmic functions. We now consider the graphs of the common and natural logarithmic functions and their geometric transformations. To understand the graphs of = log and = ln, we can compare each to the graph of its inverse, = 0 and = e, respectivel. Figure 3.23a shows that the graphs of = ln and = e are reflections of each other across the line =. Similarl, Figure 3.23b shows that the graphs of = log and = 0 are reflections of each other across this same line. = = 4 4 = e 4 = ln = 0 4 = log = ln = log (a) FIGURE 3.23 Two pairs of inverse functions. (b) [, 5] b [ 2, 2] FIGURE 3.24 The graphs of the common and natural logarithmic functions. From Figure 3.24 we can see that the graphs of = log and = ln have much in common. Figure 3.24 also shows how the differ. The geometric transformations studied in Section.5, together with our knowledge of the graphs of = ln and = log, allow us to predict the graphs of the functions in Eample 8.
6 SECTION 3.3 Logarithmic Functions and Their Graphs 279 EXAMPLE 8 Transforming Logarithmic Graphs Describe how to transform the graph of = ln or = log into the graph of the given function. (a) g2 = ln + 22 (b) h2 = ln 3-2 (c) g2 = 3 log (d) h2 = + log SOLUTION (a) The graph of g2 = ln + 22 is obtained b translating the graph of = ln 2 2 units to the left. See Figure 3.25a. (b) h2 = ln 3-2 = ln So we obtain the graph of h2 = ln 3-2 from the graph of = ln b appling, in order, a reflection across the -ais followed b a translation 3 units to the right. See Figure 3.25b. [ 3, 6] b [ 3, 3] (a) [ 3, 6] b [ 3, 3] (b) [ 3, 6] b [ 3, 3] (c) [ 3, 6] b [ 3, 3] (d) FIGURE 3.25 Transforming = ln to obtain (a) g2 = ln + 22 and (b) h2 = ln 3-2; and = log to obtain (c) g2 = 3 log and (d) h2 = + log. (Eample 8) (c) The graph of g2 = 3 log is obtained b verticall stretching the graph of ƒ2 = log b a factor of 3. See Figure 3.25c. (d) We can obtain the graph of h2 = + log from the graph of ƒ2 = log b a translation unit up. See Figure 3.25d. Now tr Eercise 4.
7 280 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions Sound Intensit Sound intensit is the energ per unit time of a sound wave over a given area, and is measured in watts per square meter W/m 2 2. Measuring Sound Using Decibels Table 3.7 lists assorted sounds. Notice that a jet at takeoff is 00 trillion times as loud as a soft whisper. Because the range of audible sound intensities is so great, common logarithms (powers of 0) are used to compare how loud sounds are. Table 3.7 Approimate Intensities of Selected Sounds Intensit Sound W/m 2 2 Hearing threshold 0-2 Soft whisper at 5 m 0 - Cit traffic 0-5 Subwa train 0-2 Pain threshold 0 0 Jet at takeoff 0 3 Source: Adapted from R. W. Reading, Eploring Phsics: Concepts and Applications. Belmont, CA: Wadsworth, 984. Bel Is for Bell The original unit for sound intensit level was the bel (B), which proved to be inconvenientl large. So the decibel, one-tenth of a bel, has replaced it. The bel was named in honor of Scottish-born American Aleander Graham Bell ( ), inventor of the telephone. DEFINITION Decibels The level of sound intensit in decibels (db) is b = 0 logi/i 0 2, where b (beta) is the number of decibels, I is the sound intensit in W/m 2, and I 0 = 0-2 W/m 2 is the threshold of human hearing (the quietest audible sound intensit). Chapter Opener Problem (from page 25) Problem: How loud is a train inside a subwa tunnel? Solution: Based on the data in Table 3.7, b = 0 logi/i 0 2 = 0 log0-2 /0-2 2 = 0 log0 0 2 = 0 # 0 = 00 So the sound intensit level inside the subwa tunnel is 00 db. QUICK REVIEW 3.3 (For help, go to Sections P. and A..) Eercise numbers with a gra background indicate problems that the authors have designed to be solved without a calculator. In Eercises 6, evaluate the epression without using a calculator In Eercises 7 0, rewrite as a base raised to a rational number eponent e 23 e 2
8 SECTION 3.3 Logarithmic Functions and Their Graphs 28 SECTION 3.3 EXERCISES In Eercises 8, evaluate the logarithmic epression without using a calculator.. log log 6 3. log log log log log log 0, log 00, log 0-4. log log ln e 3 4. ln e ln e 6. ln 7. ln 24 e 8. ln 2e 7 In Eercises 9 24, evaluate the epression without using a calculator log log log log4 23. e ln e ln/52 In Eercises 25 32, use a calculator to evaluate the logarithmic epression if it is defined, and check our result b evaluating the corresponding eponential epression. 25. log log log log ln ln ln ln In Eercises 33 36, solve the equation b changing it to eponential form. 33. log = log = log = log = -3 In Eercises 37 40, match the function with its graph. 37. ƒ2 = log ƒ2 = log ƒ2 = -ln ƒ2 = -ln 4-2 In Eercises 4 46, describe how to transform the graph of = ln into the graph of the given function. Sketch the graph b hand and support our sketch with a grapher. 4. ƒ2 = ln ƒ2 = ln ƒ2 = ln ƒ2 = ln ƒ2 = ln ƒ2 = ln 5-2 In Eercises 47 52, describe how to transform the graph of = log into the graph of the given function. Sketch the graph b hand and support with a grapher. 47. ƒ2 = - + log ƒ2 = log ƒ2 = -2 log ƒ2 = -3 log ƒ2 = 2 log ƒ2 = -3 log In Eercises 53 58, graph the function, and analze it for domain, range, continuit, increasing or decreasing behavior, boundedness, etrema, smmetr, asmptotes, and end behavior. 53. ƒ2 = log ƒ2 = ln ƒ2 = -ln ƒ2 = -log ƒ2 = 3 log ƒ2 = 5 ln Sound Intensit Use the data in Table 3.7 to compute the sound intensit in decibels for (a) a soft whisper, (b) cit traffic, and (c) a jet at takeoff. 60. Light Absorption The Beer- Lambert Law of Absorption applied to Lake Erie states that the light intensit I (in lumens), at a depth of feet, satisfies the equation log I 2 = Find the intensit of the light at a depth of 30 ft. 6. Population Growth Using the data in Table 3.8, compute a logarithmic regression model, and use it to predict when the population of San Antonio will be,500,000. (a) (c) (b) (d) Table 3.8 Population of San Antonio Year Population , , , ,5,305 Source: World Alamanac and Book of Facts 2005.
9 282 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 62. Population Deca Using the data in Table 3.9, compute a logarithmic regression model, and use it to predict when the population of Milwaukee will be 500,000. Table 3.9 Population of Milwaukee Year Population , , , ,974 Source: World Alamanac and Book of Facts Standardized Test Questions 63. True or False A logarithmic function is the inverse of an eponential function. Justif our answer. 64. True or False Common logarithms are logarithms with base 0. Justif our answer. In Eercises 65 68, ou ma use a graphing calculator to solve the problem. 65. Multiple Choice What is the approimate value of the common log of 2? (A) (B) (C) (D) (E) Multiple Choice Which statement is false? (A) log 5 = 2.5 log 2 (B) log 5 = - log 2 (C) log 5 7 log 2 (D) log 5 6 log 0 (E) log 5 = log 0 - log Multiple Choice Which statement is false about ƒ2 = ln? (A) It is increasing on its domain. (B) It is smmetric about the origin. (C) It is continuous on its domain. (D) It is unbounded. (E) It has a vertical asmptote. 68. Multiple Choice Which of the following is the inverse of ƒ2 = 2 # 3? (A) f - 2 = log (B) f - 3 /22 2 = log 2 /32 (C) f - 2 = 2 log (D) f = 3 log 2 2 (E) f - 2 = 0.5 log 3 2 Eplorations 69. Writing to Learn Parametric Graphing In the manner of Eploration, make tables and graphs for ƒ2 = 3 and its inverse ƒ - 2 = log 3. Write a comparative analsis of the two functions regarding domain, range, intercepts, and asmptotes. 70. Writing to Learn Parametric Graphing In the manner of Eploration, make tables and graphs for ƒ2 = 5 and its inverse ƒ - 2 = log 5. Write a comparative analsis of the two functions regarding domain, range, intercepts, and asmptotes. 7. Group Activit Parametric Graphing In the manner of Eploration, find the number b 7 such that the graphs of ƒ2 = b and its inverse ƒ - 2 = log b have eactl one point of intersection. What is the one point that is in common to the two graphs? 72. Writing to Learn Eplain wh zero is not in the domain of the logarithmic functions ƒ2 = log 3 and g2 = log 5. Etending the Ideas 73. Describe how to transform the graph of ƒ2 = ln into the graph of g2 = log /e. 74. Describe how to transform the graph of ƒ2 = log into the graph of g2 = log 0..
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