Logarithmic Functions. 4. f(f -1 (x)) = x and f -1 (f(x)) = x. 5. The graph of f -1 is the reflection of the graph of f about the line y = x.

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1 SECTION. Logarithmic Functions 83 SECTION. Logarithmic Functions Objectives Change from logarithmic to eponential form. Change from eponential to logarithmic form. 3 Evaluate logarithms. 4 Use basic logarithmic properties. 5 Graph logarithmic functions. 6 Find the domain of a logarithmic function. 7 Use common logarithms. 8 Use natural logarithms. The earthquake that ripped through northern California on October 7, 989, measured 7. on the Richter scale, killed more than 60 people, and injured more than 400. Shown here is San Francisco s Marina district, where shock waves tossed houses off their foundations and into the street. A higher measure on the Richter scale is more devastating than it seems because for each increase in one unit on the scale, there is a tenfold increase in the intensity of an earthquake. In this section, our focus is on the inverse of the eponential function, called the logarithmic function. The logarithmic function will help you to understand diverse phenomena, including earthquake intensity, human memory, and the pace of life in large cities. Study Tip The discussion that follows is based on our work with inverse functions in Section 8.4. Here is a summary of what you should already know about functions and their inverses.. Only one-to-one functions have inverses that are functions. A function, f, has an inverse function, f -, if there is no horizontal line that intersects the graph of f at more than one point.. If a function is one-to-one, its inverse function can be found by interchanging and y in the function s equation and solving for y. 3. If f(a) = b, then f - (b) = a. The domain of f is the range of f -. The range of f is the domain of f f(f - ()) = and f - (f()) =. 5. The graph of f - is the reflection of the graph of f about the line y =. The Definition of Logarithmic Functions No horizontal line can be drawn that intersects the graph of an eponential function at more than one point. This means that the eponential function is one-to-one and has an inverse. Let s use our switch-and-solve strategy from Section 8.4 to find the inverse. All eponential functions have inverse functions. f()=b Step. Replace f() with y: y = b. Step. Interchange and y: = b y. Step 3. Solve for y:? The question mark indicates that we do not have a method for solving b y = for y. To isolate the eponent y, a new notation, called logarithmic notation, is needed. This notation gives us a way to name the inverse of f = b. The inverse function of the eponential function with base b is called the logarithmic function with base b.

2 83 CHAPTER Eponential and Logarithmic Functions Definition of the Logarithmic Function For 7 0 and b 7 0, b Z, y = log b is equivalent to b y =. The function f = log b is the logarithmic function with base b. The equations y = log b and b y = are different ways of epressing the same thing. The first equation is in logarithmic form and the second equivalent equation is in eponential form. Notice that a logarithm, y, is an eponent. Logarithmic form allows us to isolate this eponent. You should learn the location of the base and eponent in each form. Location of Base and Eponent in Eponential and Logarithmic Forms Eponent Eponent Logarithmic Form: y=log b Base Eponential Form: b y = Base Study Tip To change from logarithmic form to the more familiar eponential form, use this pattern: y=log b means b y =. Change from logarithmic to eponential form. EXAMPLE Changing from Logarithmic to Eponential Form Write each equation in its equivalent eponential form: = log 5 3 = log b 64 a. b. c. log 3 7 = y. Solution We use the fact that y = log means b y b =. a. =log 5 means 5 =. Logarithms are eponents. b. 3=log b 64 means b 3 =64. Logarithms are eponents. c. log or means 3 y 3 7 = y y = log 3 7 = 7. CHECK POINT Write each equation in its equivalent eponential form: a. 3 = log 7 b. = log b 5 c. log 4 6 = y.

3 SECTION. Logarithmic Functions 833 Change from eponential to logarithmic form. EXAMPLE Changing from Eponential to Logarithmic Form Write each equation in its equivalent logarithmic form: = b 3 = 8 a. b. c. e y = 9. Solution We use the fact that b y = means y = log b. In logarithmic form, the eponent is isolated on one side of the equal sign. a. = means =log. b. b 3 =8 means 3=log b 8. Eponents are logarithms. Eponents are logarithms. c. e y = 9 means y = log e 9. CHECK POINT Write each equation in its equivalent logarithmic form: a. 5 = b. b 3 = 7 c. e y = Evaluate logarithms. Remembering that logarithms are eponents makes it possible to evaluate some logarithms by inspection. The logarithm of with base b, log b, is the eponent to which b must be raised to get. For eample, suppose we want to evaluate log 3. We ask, to what power gives 3? Because 5 = 3, we can conclude that log 3 = 5. EXAMPLE 3 Evaluate: log 6 a. b. c. Evaluating Logarithms log 3 9 log 5 5. Solution Logarithmic Question Needed Logarithmic Epression Epression for Evaluation Evaluated a. log 6 to what power gives 6? log because 4 6 = 4 = 6. b. log to what power gives 9? log because = = 9. c. log 5 to what power gives 5? log because = 5 5 = 5 = 5. CHECK POINT 3 Evaluate: log 0 00 log 3 3 a. b. c. log Use basic logarithmic properties. Basic Logarithmic Properties Because logarithms are eponents, they have properties that can be verified using the properties of eponents. Basic Logarithmic Properties Involving. log b b = because is the eponent to which b must be raised to obtain b. b = b. log b = 0 because 0 is the eponent to which b must be raised to obtain. b 0 =

4 834 CHAPTER Eponential and Logarithmic Functions EXAMPLE 4 Evaluate: log 7 7 a. b. log 5. Using Properties of Logarithms Solution a. Because log b b =, we conclude log 7 7 =. b. Because log b = 0, we conclude log 5 = 0. CHECK POINT 4 Evaluate: a. log 9 9 b. log 8. Now that we are familiar with logarithmic notation, let s resume and finish the switch-and-solve strategy for finding the inverse of f = b. Step. Replace f() with y: y = b. Step. Interchange and y: = b y. Step 3. Solve for y: y = log b. Step 4. Replace y with f (): f - = log b. The completed switch-and-solve strategy illustrates that if f = b, then f - = log b. The inverse of an eponential function is the logarithmic function with the same base. In Section 8.4, we saw how inverse functions undo one another. In particular, ff - = and f - f =. Applying these relationships to eponential and logarithmic functions, we obtain the following inverse properties of logarithms: Inverse Properties of Logarithms For b 7 0 and b Z, log b b = b log b = The logarithm with base b of b raised to a power equals that power. b raised to the logarithm with base b of a number equals that number. EXAMPLE 5 Evaluate: log a. b. Using Inverse Properties of Logarithms Solution a. Because log we conclude log b b =, = 5. b. Because we conclude 6 log 6 b log b =, 9 = 9. CHECK POINT 5 Evaluate: log a. b. 6 log log Graph logarithmic functions. Graphs of Logarithmic Functions How do we graph logarithmic functions? We use the fact that a logarithmic function is the inverse of an eponential function. This means that the logarithmic function reverses the coordinates of the eponential function. It also means that the graph of the logarithmic function is a reflection of the graph of the eponential function about the line y =.

5 SECTION. Logarithmic Functions 835 y f() = y = g() = log EXAMPLE 6 Graphs of Eponential and Logarithmic Functions Graph f = and g = log in the same rectangular coordinate system. Solution We first set up a table of coordinates for f =. Reversing these coordinates gives the coordinates for the inverse function g = log. 0 3 f () 4 8 g() log 4 Reverse coordinates. We now plot the ordered pairs from each table, connecting them with smooth curves. Figure.9 shows the graphs of f = and its inverse function g = log. The graph of the inverse can also be drawn by reflecting the graph of f = about the line y = FIGURE.9 The graphs of f = and its inverse function Study Tip You can obtain a partial table of coordinates for g = log without having to obtain and reverse coordinates for f =. Because g = log means g =, we begin with values for g and compute corresponding values for : Use = g() to compute. For eample, if g() =, = = = 4. Start with values for g(). 4 g() log CHECK POINT 6 Graph f = 3 and g = log 3 in the same rectangular coordinate system. Figure.0 illustrates the relationship between the graph of an eponential function, shown in blue, and its inverse, a logarithmic function, shown in red, for bases greater than and for bases between 0 and. Also shown and labeled are the eponential function s horizontal asymptote y = 0 and the logarithmic function s vertical asymptote = 0. y y = y y = (0, ) Horizontal asymptote: y = 0 Vertical asymptote: = 0 FIGURE.0 f() = b f() = b (0, ) (, 0) f () = log b Graphs of eponential and logarithmic functions Horizontal asymptote: y = 0 Vertical asymptote: b > = 0 0 < b < (, 0) f () = log b

6 836 CHAPTER Eponential and Logarithmic Functions The red graphs in Figure.0 illustrate the following general characteristics of logarithmic functions: f() = b Vertical asymptote: = 0 (0, ) y b > FIGURE.0 for b 7 ) (, 0) f () = log b y = Horizontal asymptote: y = 0 (partly repeated Characteristics of Logarithmic Functions of the Form f () log b. The domain of f = log b consists of all positive real numbers: 0, q. The range of f = log b consists of all real numbers: - q, q.. The graphs of all logarithmic functions of the form f = log b pass through the point (, 0) because f = log b = 0. The -intercept is. There is no y-intercept. 3. If b 7, f = log b has a graph that goes up to the right and is an increasing function. 4. If 0 6 b 6, f = log b has a graph that goes down to the right and is a decreasing function. (Turn back a page and verify this observation in the right portion of Figure.0.) 5. The graph of f = log b approaches, but does not touch, the y-ais. The y-ais, or = 0, is a vertical asymptote. 6 Find the domain of a logarithmic function. = y g() = log 4 ( + 3) 3 FIGURE. The domain of g = log is -3, q. The Domain of a Logarithmic Function In Section., we learned that the domain of an eponential function of the form f = b includes all real numbers and its range is the set of positive real numbers. Because the logarithmic function reverses the domain and the range of the eponential function, the domain of a logarithmic function of the form f() log b is the set of all positive real numbers. Thus, log 8 is defined because the value of in the logarithmic epression, 8, is greater than zero and therefore is included in the domain of the logarithmic function f = log. However, log 0 and log -8 are not defined because 0 and -8 are not positive real numbers and therefore are ecluded from the domain of the logarithmic function f = log. In general, the domain of f() log b g() consists of all for which g()>0. EXAMPLE 7 Finding the Domain of a Logarithmic Function Find the domain of g = log Solution The domain of g consists of all for which Solving this inequality for, we obtain 7-3. Thus, the domain of g is 5 ƒ 7-36 or -3, q. This is illustrated in Figure.. The vertical asymptote is = -3 and all points on the graph of g have -coordinates that are greater than -3. CHECK POINT 7 Find the domain of h = log Use common logarithms. Common Logarithms The logarithmic function with base 0 is called the common logarithmic function.the function f = log 0 is usually epressed as f = log. A calculator with a LOG key can be used to evaluate common logarithms. On the net page are some eamples.

7 SECTION. Logarithmic Functions 837 Most Scientific Most Graphing Display (or Approimate Logarithm Calculator Keystrokes Calculator Keystrokes Display) log log 5 LOG 5, LOG LOG ENTER LOG 5, ENTER log 5 log 5 LOG, LOG = LOG 5, LOG ENTER.393 log /- LOG LOG - 3 ENTER ERROR Some graphing calculators display an open parenthesis when the LOG key is pressed. In this case, remember to close the set of parentheses after entering the function s domain value: LOG 5 ) LOG ) ENTER. NONREAL ANS The error message or message given by many calculators for log-3 is a reminder that the domain of the common logarithmic function, f = log, is the set of positive real numbers. In general, the domain of f = log g consists of all for which g 7 0. Many real-life phenomena start with rapid growth and then the growth begins to level off. This type of behavior can be modeled by logarithmic functions. EXAMPLE 8 Modeling Height of Children The percentage of adult height attained by a boy who is years old can be modeled by f = log +, where represents the boy s age and f represents the percentage of his adult height. Approimately what percentage of his adult height has a boy attained at age eight? Solution We substitute the boy s age, 8, for and evaluate the function at 8. f = log + This is the given function. f8 = log8 + Substitute 8 for. = log 9 Graphing calculator keystrokes: LOG 9 ENTER L 76 Thus, an 8-year-old boy has attained approimately 76% of his adult height. CHECK POINT 8 Use the function in Eample 8 to answer this question: Approimately what percentage of his adult height has a boy attained at age ten? The basic properties of logarithms that were listed earlier in this section can be applied to common logarithms. Properties of Common Logarithms General Properties Common Logarithms log b = 0.. log b b =.. log b b = 3. Inverse 3. properties log = 0 log 0 = log 0 = 4. b log b = 4. 0 log =

8 838 CHAPTER Eponential and Logarithmic Functions The property log 0 = can be used to evaluate common logarithms involving powers of 0. For eample, log 00 = log 0 =, log 000 = log 0 3 = 3, and log 0 7. = 7.. EXAMPLE 9 Earthquake Intensity The magnitude, R, on the Richter scale of an earthquake of intensity I is given by R = log I, I 0 where I 0 is the intensity of a barely felt zero-level earthquake. The earthquake that destroyed San Francisco in 906 was times as intense as a zero-level earthquake. What was its magnitude on the Richter scale? Solution Because the earthquake was times as intense as a zero-level earthquake, the intensity, I, is I 0. R = log I I 0 This is the formula for magnitude on the Richter scale. R = log 08.3 I 0 I 0 = log = 8.3 Substitute I 0 for I. Simplify. Use the property log 0. San Francisco s 906 earthquake registered 8.3 on the Richter scale. CHECK POINT 9 Use the formula in Eample 9 to solve this problem. If an earthquake is 0,000 times as intense as a zero-level quake I = 0,000I 0, what is its magnitude on the Richter scale? 8 Use natural logarithms. Natural Logarithms The logarithmic function with base e is called the natural logarithmic function. The function f = log e is usually epressed as f = ln, read el en of. A calculator with an LN key can be used to evaluate natural logarithms. Keystrokes are identical to those shown for common logarithmic evaluations on page 837. Like the domain of all logarithmic functions, the domain of the natural logarithmic function f = ln is the set of all positive real numbers. Thus, the domain of f = ln g consists of all for which g 7 0. EXAMPLE 0 Find the domain of each function: f = ln3 - a. b. Finding Domains of Natural Logarithmic Functions g = ln - 3. [ 0, 0, ] by [ 0, 0, ] FIGURE. The domain of f = ln3 - is - q, 3. Solution a. The domain of f = ln3 - consists of all for which Solving this inequality for, we obtain 6 3. Thus, the domain of f is 5 ƒ 6 36 or - q, 3. This is verified by the graph in Figure..

9 SECTION. Logarithmic Functions 839 b. The domain of g = ln - 3 consists of all for which It follows that the domain of g is all real numbers ecept 3.Thus, the domain of g is 5 ƒ Z 36 or - q, 3 3, q. This is shown by the graph in Figure.3. To make it more obvious that 3 is ecluded from the domain, we used a DOT format. [ 0, 0, ] by [ 0, 0, ] FIGURE.3 3 is ecluded from the domain of g() = ln( - 3). a. b. CHECK POINT 0 f = ln4 - g = ln. Find the domain of each function: The basic properties of logarithms that were listed earlier in this section can be applied to natural logarithms. Properties of Natural Logarithms General Natural Properties Logarithms log b = 0.. log b b =.. log b b = 3. Inverse 3. properties ln = 0 ln e = ln e = 4. b log b = 4. e ln = Eamine the inverse properties, ln e = and e ln =. Can you see how ln and e undo one another? For eample, EXAMPLE ln e =, ln e 7 = 7, e ln =, and e ln 7 = 7. Dangerous Heat: Temperature in an Enclosed Vehicle When the outside air temperature is anywhere from 7 to 96 Fahrenheit, the temperature in an enclosed vehicle climbs by 43 in the first hour.the bar graph in Figure.4 shows the temperature increase throughout the hour. The function f = 3.4 ln -.6 models the temperature increase, f, in degrees Fahrenheit, after minutes. Use the function to find the temperature increase, to the nearest degree, after 50 minutes. How well does the function model the actual increase shown in Figure.4? Temperature Increase in an Enclosed Vehicle Temperature Increase ( F) Minutes FIGURE.4 Source: Professor Jan Null, San Francisco State University

10 840 CHAPTER Eponential and Logarithmic Functions Temperature Increase ( F) FIGURE.4 Temperature Increase in an Enclosed Vehicle Minutes 4 50 (repeated) Solution We find the temperature increase after 50 minutes by substituting 50 for and evaluating the function at 50. f = 3.4 ln -.6 f50 = 3.4 ln L 4 This is the given function. Substitute 50 for. Graphing calculator keystrokes: 3.4 ln ENTER. On some calculators, a parenthesis is needed after 50. According to the function, the temperature will increase by approimately 4 after 50 minutes. Because the increase shown in Figure.4 is 4, the function models the actual increase etremely well. CHECK POINT Use the function in Eample to find the temperature increase, to the nearest degree, after 30 minutes. How well does the function model the actual increase shown in Figure.4?. EXERCISE SET Practice Eercises In Eercises 8, write each equation in its equivalent eponential form.. 4 = log 6. 6 = log = log 3 4. = log = log b = log b 7 7. log 6 6 = y 8. log 5 5 = y In Eercises 9 0, write each equation in its equivalent logarithmic form = = =. 5-3 = = = = 6. 5 = 7. b 3 = b 3 = y = y = 300 In Eercises 4, evaluate each epression without using a calculator.. log 4 6. log log log log log log 8 8. log log log log 3. log log log log log 37. log log log 40. log log7 3 8 log Graph f = 4 and g = log 4 in the same rectangular coordinate system. 44. Graph f = 5 and g = log 5 in the same rectangular coordinate system. 45. Graph f = A B and g = log in the same rectangular coordinate system. 46. Graph f = A 4 B and g = log in the same 4 rectangular coordinate system. In Eercises 47 5, find the domain of each logarithmic function. 47. f = log f = log f = log f = log7-5. f = ln - 5. f = ln - 7 In Eercises 53 66, evaluate each epression without using a calculator. 53. log log log log log 33 log ln 60. ln e 6. ln e 6 6. ln e ln e ln e 7 ln e ln e

11 SECTION. Logarithmic Functions 84 In Eercises 67 7, simplify each epression. d. 67. ln e e 3 ln ln e 70. e ln log 7. 0 log 3 Practice PLUS In Eercises 73 76, write each equation in its equivalent eponential form. Then solve for. e. 73. log 3 - = 74. log = 75. log 4 = log 64 = 3 In Eercises 77 80, evaluate each epression without using a calculator. f. 77. log 3 log log 5 log log log logln e In Eercises 8 86, match each function with its graph. The graphs are labeled (a) through (f), and each graph is displayed in a [-5, 5, ] by [-5, 5, ] viewing rectangle. 8. f = ln + 8. f = ln f = ln f = ln f = ln f = ln - a. Application Eercises The percentage of adult height attained by a girl who is years old can be modeled by f = log - 4, where represents the girl s age (from 5 to 5) and f represents the percentage of her adult height. Use the formula to solve Eercises Round answers to the nearest tenth of a percent. 87. Approimately what percentage of her adult height has a girl attained at age 3? 88. Approimately what percentage of her adult height has a girl attained at age ten? b. c. The bar graph indicates that the percentage of first-year college students epressing antifeminist views declined after 970. Use this information to solve Eercises Percentage Agreeing with the Statement 70% 60% 50% 40% 30% 0% 0% Opposition to Feminism among First-Year United States College Students, Statement: The activities of married women are best confined to the home and family Year Source: John Macionis, Sociology, Eleventh Edition, Prentice Hall, 007 Men Women

12 84 CHAPTER Eponential and Logarithmic Functions 89. The function f = ln + 53 models the percentage of first-year college men, f, epressing antifeminist views (by agreeing with the statement) years after 969. a. Use the function to find the percentage of first-year college men epressing antifeminist views in 004. Round to one decimal place. Does this function value overestimate or underestimate the percentage displayed by the graph on the previous page? By how much? 93. Students in a psychology class took a final eamination. As part of an eperiment to see how much of the course content they remembered over time, they took equivalent forms of the eam in monthly intervals thereafter. The average score for the group, ft, after t months was modeled by the function ft = 88-5 lnt +, 0 t. a. What was the average score on the original eam? b. What was the average score, to the nearest tenth, after months? 4 months? 6 months? 8 months? 0 months? one year? b. Use the function to project the percentage of first-year college men who will epress antifeminist views in 00. Round to one decimal place. 90. The function c. Sketch the graph of f (either by hand or with a graphing utility). Describe what the graph indicates in terms of the material retained by the students. f = ln models the percentage of first-year college women, f, epressing antifeminist views (by agreeing with the statement) years after 969. a. Use the function to find the percentage of first-year college women epressing antifeminist views in 004. Round to one decimal place. Does this function value overestimate or underestimate the percentage displayed by the graph on the previous page? By how much? b. Use the function to project the percentage of first-year college women who will epress antifeminist views in 00. Round to one decimal place. The loudness level of a sound, D, in decibels, is given by the formula D = 0 log0 I, where I is the intensity of the sound, in watts per meter. Decibel levels range from 0, a barely audible sound, to 60, a sound resulting in a ruptured eardrum. Use the formula to solve Eercises The sound of a blue whale can be heard 500 miles away, reaching an intensity of 6.3 * 0 6 watts per meter. Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum? 9. What is the decibel level of a normal conversation, 3. * 0-6 watt per meter? Writing in Mathematics 94. Describe the relationship between an equation in logarithmic form and an equivalent equation in eponential form. 95. What question can be asked to help evaluate log 3 8? 96. Eplain why the logarithm of with base b is Describe the following property using words: log b b =. 98. Eplain how to use the graph of f = to obtain the graph of g = log. 99. Eplain how to find the domain of a logarithmic function. 00. Logarithmic models are well suited to phenomena in which growth is initially rapid, but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function. 0. Suppose that a girl is 4 feet 6 inches at age 0. Eplain how to use the function in Eercises to determine how tall she can epect to be as an adult. Technology Eercises In Eercises 0 05, graph f and g in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of f. 0. f = ln, g = ln f = ln, g = ln f = log, g = -log 05. f = log, g = log - +

13 SECTION. Logarithmic Functions Students in a mathematics class took a final eamination. They took equivalent forms of the eam in monthly intervals thereafter. The average score, ft, for the group after t months is modeled by the human memory function ft = 75-0 logt +, where 0 t. Use a graphing utility to graph the function. Then determine how many months will elapse before the average score falls below In parts a c, graph f and g in the same viewing rectangle. a. f = ln3, g = ln 3 + ln b. f = log5, g = log 5 + log c. f = ln 3, g = ln + ln 3 d. Describe what you observe in parts (a) (c). Generalize this observation by writing an equivalent epression for log b MN, where M 7 0 and N 7 0. e. Complete this statement: The logarithm of a product is equal to. 08. Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. y =, y =, y = e, y = ln, y =, y = 7. Without using a calculator, find the eact value of 8. Without using a calculator, find the eact value of log 4 [log 3 log 8]. 9. Without using a calculator, determine which is the greater number: log 4 60 or log Review Eercises 0. Solve the system: (Section 4.3, Eample 4). Factor completely: (Section 6.5, Eample 8) log log p log 8 - log = - 5y 3 - y = y + y.. Solve: or (Section 9., Eample 6) Critical Thinking Eercises Make Sense? In Eercises 09, determine whether each statement makes sense or does not make sense and eplain your reasoning. 09. I estimate that log lies between and because = 8 and 8 = When graphing a logarithmic function, I like to show the graph of its horizontal asymptote.. I can evaluate some common logarithms without having to use a calculator.. An earthquake of magnitude 8 on the Richter scale is twice as intense as an earthquake of magnitude 4. In Eercises 3 6, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. log 8 3. log 4 = log- 00 = The domain of f = log is - q, q. 6. log b is the eponent to which b must be raised to obtain. Preview Eercises Eercises 3 5 will help you prepare for the material covered in the net section. In each eercise, evaluate the indicated logarithmic epressions without using a calculator. 3. a. Evaluate: b. Evaluate: c. What can you conclude about log 3, or log 8 # 4? 4. a. Evaluate: b. Evaluate: c. What can you conclude about 5. a. Evaluate: b. Evaluate: log 3. log 8 + log 4. log 6. log 3 - log. log 6, or log a 3 b? log 3 8. log 3 9. c. What can you conclude about log 3 8, or log 3 9?