3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS

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Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a.

1 Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions. Recognize, evaluate, and graph natural logarithmic functions. Use logarithmic functions to model and solve real-life problems. Wh ou should learn it Logarithmic functions are often used to model scientific observations. For instance, in Eercise 97 on page 6, a logarithmic function is used to model human memor. Logarithmic Functions In Section.9, ou studied the concept of an inverse function. There, ou learned that if a function is one-to-one that is, if the function has the propert that no horizontal line intersects the graph of the function more than once the function must have an inverse function. B looking back at the graphs of the eponential functions introduced in Section., ou will see that ever function of the form f a passes the Horizontal Line Test and therefore must have an inverse function. This inverse function is called the logarithmic function with base a. Definition of Logarithmic Function with Base a For > 0, a > 0, and a, log if and onl if a a. The function given b f log a Read as log base a of. is called the logarithmic function with base a. The equations log and a a are equivalent. The first equation is in logarithmic form and the second is in eponential form. For eample, the logarithmic equation log 9 can be rewritten in eponential form as 9. The eponential equation 5 5 can be rewritten in logarithmic form as log 5 5. When evaluating logarithms, remember that a logarithm is an eponent. This means that log a is the eponent to which a must be raised to obtain. For instance, log 8 because must be raised to the third power to get 8. Eample Evaluating Logarithms Use the definition of logarithmic function to evaluate each logarithm at the indicated value of. a. f log, b. f log, c. f log 4, d. f log 0, 00 a. f log 5 because 5. b. f log 0 because 0. c. f log 4 because 4 4. d. f 00 log 0 00 because 0 0 Now tr Eercise. 00.

2 8 Chapter Eponential and Logarithmic Functions The logarithmic function with base 0 is called the common logarithmic function. It is denoted b log 0 or simpl b log. On most calculators, this function is denoted b LOG. Eample shows how to use a calculator to evaluate common logarithmic functions. You will learn how to use a calculator to calculate logarithms to an base in the net section. Eample Evaluating Common Logarithms on a Calculator Use a calculator to evaluate the function given b f log at each value of. a. 0 b. c..5 d. Function Value Graphing Calculator Kestrokes Displa a. f 0 log 0 LOG 0 ENTER b. LOG f log ENTER c. f.5 log.5 LOG.5 ENTER d. f log LOG ENTER ERROR Note that the calculator displas an error message (or a comple number) when ou tr to evaluate log. The reason for this is that there is no real number power to which 0 can be raised to obtain. Now tr Eercise 9. The following properties follow directl from the definition of the logarithmic function with base a. Properties of Logarithms. log because a 0 a 0.. log because a a a a.. log and a log a a a Inverse Properties 4. If log a log a, then. One-to-One Propert Eample Using Properties of Logarithms a. Simplif: log 4 b. Simplif: log 7 7 c. Simplif: 6 log 6 0 a. Using Propert, it follows that log 4 0. b. Using Propert, ou can conclude that log 7 7. c. Using the Inverse Propert (Propert ), it follows that 6 log Now tr Eercise. You can use the One-to-One Propert (Propert 4) to solve simple logarithmic equations, as shown in Eample 4.

3 Section. Logarithmic Functions and Their Graphs 9 Eample 4 Using the One-to-One Propert a. log log Original equation One-to-One Propert b. log log c. log 4 6 log ±4 Now tr Eercise 85. Graphs of Logarithmic Functions To sketch the graph of log a, ou can use the fact that the graphs of inverse functions are reflections of each other in the line. Eample 5 Graphs of Eponential and Logarithmic Functions FIGURE.4 f() = = g() = log In the same coordinate plane, sketch the graph of each function. a. f b. g log a. For f, construct a table of values. B plotting these points and connecting them with a smooth curve, ou obtain the graph shown in Figure.4. 0 f b. Because g log is the inverse function of f, the graph of g is obtained b plotting the points f, and connecting them with a smooth curve. The graph of g is a reflection of the graph of f in the line, as shown in Figure.4. Now tr Eercise Vertical asmptote: = 0 f() = log FIGURE.5 Eample 6 Sketching the Graph of a Logarithmic Function Sketch the graph of the common logarithmic function f log. Identif the vertical asmptote. Begin b constructing a table of values. Note that some of the values can be obtained without a calculator b using the Inverse Propert of Logarithms. Others require a calculator. Net, plot the points and connect them with a smooth curve, as shown in Figure.5. The vertical asmptote is 0 ( -ais). 00 Without calculator With calculator f log Now tr Eercise 4.

4 0 Chapter Eponential and Logarithmic Functions The nature of the graph in Figure.5 is tpical of functions of the form f log a, a >. The have one -intercept and one vertical asmptote. Notice how slowl the graph rises for >. The basic characteristics of logarithmic graphs are summarized in Figure.6. FIGURE.6 (, 0) = log a Graph of log a, a > Domain: 0, Range:, -intercept:, 0 Increasing One-to-one, therefore has an inverse function -ais is a vertical asmptote log as 0 a. Continuous Reflection of graph of a about the line The basic characteristics of the graph of f a are shown below to illustrate the inverse relation between f a and g log a. Domain:, Range: 0, -intercept: 0, -ais is a horizontal asmptote a 0 as. In the net eample, the graph of log a is used to sketch the graphs of functions of the form f b ± log a c. Notice how a horizontal shift of the graph results in a horizontal shift of the vertical asmptote. Eample 7 Shifting Graphs of Logarithmic Functions You can use our understanding of transformations to identif vertical asmptotes of logarithmic functions. For instance, in Eample 7(a), the graph of g f shifts the graph of f one unit to the right. So, the vertical asmptote of g is, one unit to the right of the vertical asmptote of the graph of f. The graph of each of the functions is similar to the graph of f log. a. Because g log f, the graph of g can be obtained b shifting the graph of f one unit to the right, as shown in Figure.7. b. Because h log f, the graph of h can be obtained b shifting the graph of f two units upward, as shown in Figure.8. f() = log (, 0) (, ) (, 0) h() = + log f() = log You can review the techniques for shifting, reflecting, and stretching graphs in Section.7. g() = log( ) FIGURE.7 FIGURE.8 Now tr Eercise 45. (, 0)

5 Section. Logarithmic Functions and Their Graphs The Natural Logarithmic Function B looking back at the graph of the natural eponential function introduced on page 0 in Section., ou will see that f e is one-to-one and so has an inverse function. This inverse function is called the natural logarithmic function and is denoted b the special smbol ln, read as the natural log of or el en of. Note that the natural logarithm is written without a base. The base is understood to be e. (, e ) (0, ) f() = e (, e) (, 0) (, ) (e, ) = g() = f () = ln Reflection of graph of f e about the line FIGURE.9 e The Natural Logarithmic Function The function defined b f log e ln, > 0 is called the natural logarithmic function. The definition above implies that the natural logarithmic function and the natural eponential function are inverse functions of each other. So, ever logarithmic equation can be written in an equivalent eponential form, and ever eponential equation can be written in logarithmic form. That is, ln and e are equivalent equations. Because the functions given b f e and g ln are inverse functions of each other, their graphs are reflections of each other in the line. This reflective propert is illustrated in Figure.9. On most calculators, the natural logarithm is denoted b LN, as illustrated in Eample 8. Eample 8 Evaluating the Natural Logarithmic Function Use a calculator to evaluate the function given b f ln for each value of. a. b. 0. c. d. WARNING / CAUTION Notice that as with ever other logarithmic function, the domain of the natural logarithmic function is the set of positive real numbers be sure ou see that ln is not defined for zero or for negative numbers. Function Value Graphing Calculator Kestrokes Displa a. f ln LN ENTER b. f 0. ln 0. LN. ENTER.0978 c. f ln LN ENTER ERROR d. f ln LN ENTER Now tr Eercise 67. In Eample 8, be sure ou see that ln gives an error message on most calculators. (Some calculators ma displa a comple number.) This occurs because the domain of ln is the set of positive real numbers (see Figure.9). So, ln is undefined. The four properties of logarithms listed on page 8 are also valid for natural logarithms.

6 Chapter Eponential and Logarithmic Functions Properties of Natural Logarithms. ln 0 because e 0.. ln e because e e.. ln e and e ln Inverse Properties 4. If ln ln, then. One-to-One Propert Eample 9 Using Properties of Natural Logarithms Use the properties of natural logarithms to simplif each epression. ln a. ln b. e c. d. ln e e ln 5 a. ln Inverse Propert b. e ln 5 5 Inverse Propert e ln e ln c. Propert d. ln e Propert 0 0 Now tr Eercise 7. Eample 0 Finding the Domains of Logarithmic Functions Find the domain of each function. a. f ln b. g ln c. h ln a. Because ln is defined onl if > 0, it follows that the domain of f is,. The graph of f is shown in Figure.0. b. Because ln is defined onl if > 0, it follows that the domain of g is,. The graph of g is shown in Figure.. c. Because ln is defined onl if > 0, it follows that the domain of h is all real numbers ecept 0. The graph of h is shown in Figure.. 4 f() = ln( ) 4 5 g() = ln( ) h() = ln FIGURE.0 FIGURE. FIGURE. Now tr Eercise 75.

7 Section. Logarithmic Functions and Their Graphs Application Eample Human Memor Model Students participating in a pscholog eperiment attended several lectures on a subject and were given an eam. Ever month for a ear after the eam, the students were retested to see how much of the material the remembered. The average scores for the group are given b the human memor model f t 75 6 ln t, 0 t, where t is the time in months. a. What was the average score on the original t 0 eam? b. What was the average score at the end of t months? c. What was the average score at the end of t 6 months? Algebraic a. The original average score was f ln 0 Substitute 0 for t ln Simplif. Propert of natural logarithms 75. b. After months, the average score was f 75 6 ln Substitute for t ln Simplif Use a calculator c. After 6 months, the average score was f ln 6 Substitute 6 for t ln 7 Simplif Use a calculator. 6.. Graphical Use a graphing utilit to graph the model 75 6 ln. Then use the value or trace feature to approimate the following. a. When 0, 75 (see Figure.). So, the original average score was 75. b. When, 68.4 (see Figure.4). So, the average score after months was about c. When 6, 6. (see Figure.5). So, the average score after 6 months was about FIGURE. FIGURE Now tr Eercise FIGURE.5 CLASSROOM DISCUSSION Analzing a Human Memor Model Use a graphing utilit to determine the time in months when the average score in Eample was 60. Eplain our method of solving the problem. Describe another wa that ou can use a graphing utilit to determine the answer.

8 4 Chapter Eponential and Logarithmic Functions. EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. The inverse function of the eponential function given b f a is called the function with base a.. The common logarithmic function has base.. The logarithmic function given b f ln is called the logarithmic function and has base. 4. The Inverse Properties of logarithms and eponentials state that log a a and. 5. The One-to-One Propert of natural logarithms states that if ln ln, then. 6. The domain of the natural logarithmic function is the set of. SKILLS AND APPLICATIONS In Eercises 7 4, write the logarithmic equation in eponential form. For eample, the eponential form of log is log log log 0. log log 4 5. log log log 8 4 In Eercises 5, write the eponential equation in logarithmic form. For eample, the logarithmic form of 8 is log In Eercises 8, evaluate the function at the indicated value of without using a calculator. Function Value f log f log 5 f log 8 f log g log a g log b a b In Eercises 9, use a calculator to evaluate f log at the indicated value of. Round our result to three decimal places In Eercises 6, use the properties of logarithms to simplif the epression.. log 7 4. log In Eercises 7 44, find the domain, -intercept, and vertical asmptote of the logarithmic function and sketch its graph. 7. f log 4 8. g log 6 9. log 40. h log 4 4. f log 6 4. log log 44. log In Eercises 45 50, use the graph of g log to match the given function with its graph. Then describe the relationship between the graphs of f and g. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) log 4 4 (b) (d) (e) (f ) log

9 Section. Logarithmic Functions and Their Graphs f log f log f log 50. f log f log f log 87. log log ln 4 ln ln ln 9. log 5 log ln 7 ln 7 ln ln 6 In Eercises 5 58, write the logarithmic equation in eponential form. 5. ln ln ln ln ln ln ln ln e In Eercises 59 66, write the eponential equation in logarithmic form. 59. e e e e e e e e In Eercises 67 70, use a calculator to evaluate the function at the indicated value of. Round our result to three decimal places. Function Value f ln f ln g 8 ln g ln In Eercises 7 74, evaluate g ln at the indicated value of without using a calculator. 7. e 5 7. e 4 7. e e 5 In Eercises 75 78, find the domain, -intercept, and vertical asmptote of the logarithmic function and sketch its graph. 75. f ln h ln g ln 78. f ln In Eercises 79 84, use a graphing utilit to graph the function. Be sure to use an appropriate viewing window. 79. f log f log 6 8. f ln 8. f ln 8. f ln f ln In Eercises 85 9, use the One-to-One Propert to solve the equation for. 85. log 5 log log log 9 9. MONTHLY PAYMENT The model t 6.65 ln 750, > 750 approimates the length of a home mortgage of $50,000 at 6% in terms of the monthl pament. In the model, t is the length of the mortgage in ears and is the monthl pament in dollars. (a) Use the model to approimate the lengths of a $50,000 mortgage at 6% when the monthl pament is $897.7 and when the monthl pament is $ (b) Approimate the total amounts paid over the term of the mortgage with a monthl pament of $897.7 and with a monthl pament of $ (c) Approimate the total interest charges for a monthl pament of $897.7 and for a monthl pament of $ (d) What is the vertical asmptote for the model? Interpret its meaning in the contet of the problem. 94. COMPOUND INTEREST A principal P, invested at 5 % and compounded continuousl, increases to an amount K times the original principal after t ears, where t is given b t ln K (a) Complete the table and interpret our results. K t (b) Sketch a graph of the function. 95. CABLE TELEVISION The numbers of cable television sstems C (in thousands) in the United States from 00 through 006 can be approimated b the model C t ln t, t 6 where t represents the ear, with t corresponding to 00. (Source: Warren Communication News) (a) Complete the table. t C (b) Use a graphing utilit to graph the function. (c) Can the model be used to predict the numbers of cable television sstems beond 006? Eplain.

10 6 Chapter Eponential and Logarithmic Functions 96. POPULATION The time t in ears for the world population to double if it is increasing at a continuous rate of r is given b t ln r. (a) Complete the table and interpret our results. r t (b) Use a graphing utilit to graph the function. 97. HUMAN MEMORY MODEL Students in a mathematics class were given an eam and then retested monthl with an equivalent eam. The average scores for the class are given b the human memor model f t 80 7 log t, 0 t, where t is the time in months. (a) Use a graphing utilit to graph the model over the specified domain. (b) What was the average score on the original eam t 0? (c) What was the average score after 4 months? (d) What was the average score after 0 months? 98. SOUND INTENSITY The relationship between the number of decibels and the intensit of a sound I in watts per square meter is 0 log I (a) Determine the number of decibels of a sound with an intensit of watt per square meter. (b) Determine the number of decibels of a sound with an intensit of 0 watt per square meter. (c) The intensit of the sound in part (a) is 00 times as great as that in part (b). Is the number of decibels 00 times as great? Eplain. EXPLORATION 0. TRUE OR FALSE? In Eercises 99 and 00, determine whether the statement is true or false. Justif our answer. 99. You can determine the graph of f log 6 b graphing g 6 and reflecting it about the -ais. 00. The graph of f log contains the point 7,. In Eercises 0 04, sketch the graphs of f and g and describe the relationship between the graphs of f and g. What is the relationship between the functions f and g? f, f 5, f e, f 8, g log g log 5 g ln g log THINK ABOUT IT Complete the table for f 0. 0 f Complete the table for f log. 00 f Compare the two tables. What is the relationship between f 0 and f log? 06. GRAPHICAL ANALYSIS Use a graphing utilit to graph f and g in the same viewing window and determine which is increasing at the greater rate as approaches. What can ou conclude about the rate of growth of the natural logarithmic function? (a) f ln, (b) f ln, g g (a) Complete the table for the function given b f ln f (b) Use the table in part (a) to determine what value f approaches as increases without bound. (c) Use a graphing utilit to confirm the result of part (b). 08. CAPSTONE The table of values was obtained b evaluating a function. Determine which of the statements ma be true and which must be false. (a) is an eponential function of. 0 (b) is a logarithmic function of. (c) is an eponential function of. (d) is a linear function of WRITING Eplain wh log a is defined onl for 0 < a < and a >. In Eercises 0 and, (a) use a graphing utilit to graph the function, (b) use the graph to determine the intervals in which the function is increasing and decreasing, and (c) approimate an relative maimum or minimum values of the function. 0. f ln. h ln

3.2 Logarithmic Functions and Their Graphs

96 Chapter 3 Eponential and Logarithmic Functions 3.2 Logarithmic Functions and Their Graphs Logarithmic Functions In Section.6, you studied the concept of an inverse function. There, you learned that