304 86. 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS 3 2 ⱍ 102. (A) For f ⫽ {(x, y) y ⫽ (13)x ⫽ 3⫺x}, graph f, f ⫺1, and y ⫽ x on the same coordinate system. logb 4 ⫺ 23 logb 8 ⫹ 2 logb 2 ⫽ logb x If logb 2 ⫽ 0.69, logb 3 ⫽ 1.10, and logb 5 ⫽ 1.61, find the value of each expression in Problems 87–96. 87. logb 30 90. logb 53 93. logb 兹2 3 logb 25 88. logb 12 89. 91. logb 27 92. logb 16 94. logb 兹3 95. logb 兹0.9 96. logb 兹1.5 3 (B) Indicate the domain and range of f and f ⫺1. (C) What other name can you use for the inverse of f ? Find the inverse of each function in Problems 103–106. 103. f(x) ⫽ 53x⫺1 ⫹ 4 104. g(x) ⫽ 32x⫺3 ⫺ 2 C 105. g(x) ⫽ 3 loge (5x ⫺ 2) In Problems 97–100, (A) Use the graph of y ⫽ log2 x (Fig. 1) and graph transformations to sketch the graph of f. ⫺1 (B) Find f and use the Draw Inverse routine on a graphing utility to check the graph in part A. 97. f(x) ⫽ log2 (x ⫺ 2) 106. f(x) ⫽ 2 ⫹ loge (5x ⫺ 3) 107. Explain why the graph of the reflection of the function y ⫽ 3x in the line y ⫽ x is not the graph of a function. 2 108. Explain why the graph of the reflection of the function y ⫽ 2ⱍxⱍ in the line y ⫽ x is not the graph of a function. 109. Write loge x ⫺ loge 100 ⫽ ⫺0.08t in an exponential form that is free of logarithms. 98. f(x) ⫽ log2 (x ⫹ 3) 99. f(x) ⫽ log2 x ⫺ 2 110. Write loge x ⫺ loge C ⫹ kt ⫽ 0 in an exponential form that is free of logarithms. 100. f(x) ⫽ log2 x ⫹ 3 ⱍ 101. (A) For f ⫽ {(x, y) y ⫽ (12)x ⫽ 2⫺x}, graph f, f ⫺1, and y ⫽ x on the same coordinate system. (B) Indicate the domain and range of f and f ⫺1. (C) What other name can you use for the inverse of f ? 111. Prove that logb (M/N) ⫽ logb M ⫺ logb N under the hypotheses of Theorem 1. 112. Prove that logb Mp ⫽ p logb M under the hypotheses of Theorem 1. Section 4-6 Common and Natural Logarithms Common and Natural Logarithmic Functions Applications John Napier (1550–1617) is credited with the invention of logarithms, which evolved out of an interest in reducing the computational strain in research in astronomy. This new computational tool was immediately accepted by the scientific world. Now, with the availability of inexpensive calculators, logarithms have lost most of their importance as a computational device. However, the logarithmic concept has been greatly generalized since its conception, and logarithmic functions are used widely in both theoretical and applied sciences. Of all possible logarithmic bases, the base e and the base 10 are used almost exclusively. Before we can use logarithms in certain practical problems, we need to be able to approximate the logarithm of any positive number to either base 10 or base e. And conversely, if we are given the logarithm of a number to base 10 or base e, we need to be able to approximate the number. Historically, tables were used for this purpose, but now calculators are used since they are faster and can find far more values than any table can possibly include. 4-6 Common and Natural Logarithms 305 Common and Natural Logarithmic Functions Common logarithms, also called Briggsian logarithms, are logarithms with base 10. Natural logarithms, also called Naperian logarithms, are logarithms with base e. Most calculators have a function key labeled “log” and a function key labeled “ln.” The former represents the common logarithmic function and the latter the natural logarithmic function. In fact, “log” and “ln” are both used extensively in mathematical literature, and whenever you see either used in this book without a base indicated, they should be interpreted as in the following box. LOGARITHMIC FUNCTIONS Explore/Discuss 1 EXAMPLE 1 Solutions y ⫽ log x ⫽ log10 x Common logarithmic function y ⫽ ln x ⫽ loge x Natural logarithmic function (A) Sketch the graph of y ⫽ 10x, y ⫽ log x, and y ⫽ x in the same coordinate system and state the domain and range of the common logarithmic function. (B) Sketch the graph of y ⫽ ex, y ⫽ ln x, and y ⫽ x in the same coordinate system and state the domain and range of the natural logarithmic function. Calculator Evaluation of Logarithms Use a calculator to evaluate each to six decimal places. (A) log 3,184 (B) ln 0.000 349 (C) log (⫺3.24) (A) log 3,184 ⫽ 3.502 973 (B) ln 0.000 349 ⫽ ⫺7.960 439 (C) log (⫺3.24) ⫽ Error Why is an error indicated in part C? Because ⫺3.24 is not in the domain of the log function. [Note: Calculators display error messages in various ways. Some calculators use a more advanced definition of logarithmic functions that involves complex numbers. They will display an ordered pair, representing a complex number, as the value of log (⫺3.24), rather than an error message. You should interpret such a display as indicating that the number entered is not in the domain of the logarithmic function as we have defined it.] MATCHED PROBLEM 1 Use a calculator to evaluate each to six decimal places. (A) log 0.013 529 (B) ln 28.693 28 (C) ln (⫺0.438) 306 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS When working with common and natural logarithms, we follow the common practice of using the equal sign “⫽” where it might be more appropriate to use the approximately equal sign “⬇.” No harm is done as long as we keep in mind that in a statement such as log 3.184 ⫽ 0.503, the number on the right is only assumed accurate to three decimal places and is not exact. Graphs of the functions f(x) ⫽ log x and g(x) ⫽ ln x are shown in the graphing utility display of Figure 1. Which graph belongs to which function? It appears from the display that one of the functions may be a constant multiple of the other. Is that true? Find and discuss the evidence for your answer. Explore/Discuss 2 FIGURE 1 2 0 5 ⫺2 EXAMPLE Calculator Evaluation of Logarithms 2 Solutions Use a calculator to evaluate each expression to three decimal places. log 2 2 (A) (B) log (C) log 2 ⫺ log 1.1 log 1.1 1.1 log 2 ⫽ 7.273 log 1.1 2 ⫽ 0.260 (B) log 1.1 (A) log 2 2 ⫽ log 2 ⫺ log 1.1, but log ⫽ log 1.1 1.1 log 2 ⫺ log 1.1 (see Theorem 1, Section 4-5). (C) log 2 ⫺ log 1.1 ⫽ 0.260. Note that MATCHED PROBLEM 2 Use a calculator to evaluate each to three decimal places. ln 3 3 (A) (B) ln (C) ln 3 ⫺ ln 1.08 ln 1.08 1.08 We now turn to the second problem: Given the logarithm of a number, find the number. To solve this problem, we make direct use of the logarithmic–exponential relationships discussed in Section 4-5. 4-6 Common and Natural Logarithms 307 –EXPONENTIAL RELATIONSHIPS LOGARITHMIC– EXAMPLE 3 Solutions log x ⫽ y is equivalent to x ⫽ 10y ln x ⫽ y is equivalent to x ⫽ ey Solving logb x ⴝ y for x Find x to three significant digits, given the indicated logarithms. (A) log x ⫽ ⫺9.315 (B) ln x ⫽ 2.386 (A) log x ⫽ ⫺9.315 x ⫽ 10⫺9.315 Change to equivalent exponential form. ⫺10 ⫽ 4.84 ⫻ 10 Notice that the answer is displayed in scientific notation in the calculator. (B) ln x ⫽ 2.386 x ⫽ e2.386 Change to equivalent exponential form. ⫽ 10.9 MATCHED PROBLEM 3 Explore/Discuss 3 Find x to four significant digits, given the indicated logarithms. (A) ln x ⫽ ⫺5.062 (B) log x ⫽ 12.0821 Example 3 was solved algebraically using the logarithmic–exponential relationships. Use the intersection routine on a graphing utility to solve this problem graphically. Discuss the relative merits of the two approaches. Applications We now consider three applications that are solved using common and natural logarithms. The first application concerns sound intensity; the second, earthquake intensity; and the third, rocket flight theory. Sound Intensity The human ear is able to hear sound over an incredible range of intensities. The loudest sound a healthy person can hear without damage to the eardrum has an intensity 1 trillion (1,000,000,000,000) times that of the softest sound a person can hear. Working directly with numbers over such a wide range is very cumbersome. Since the logarithm, with base greater than 1, of a number increases much more slowly than the number itself, logarithms are often used to create more convenient compressed scales. The decibel scale for sound intensity 308 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS is an example of such a scale. The decibel, named after the inventor of the telephone, Alexander Graham Bell (1847–1922), is defined as follows: D ⴝ 10 log I I0 Decibel scale (1) where D is the decibel level of the sound, I is the intensity of the sound measured in watts per square meter (W/m2), and I0 is the intensity of the least audible sound that an average healthy young person can hear. The latter is standardized to be I0 ⫽ 10⫺12 watt per square meter. Table 1 lists some typical sound intensities from familiar sources. T A B L E EXAMPLE 4 Solution 1 Typical Sound Intensities Sound Intensity (W/m2) Sound 1.0 ⫻ 10⫺12 5.2 ⫻ 10⫺10 3.2 ⫻ 10⫺6 8.5 ⫻ 10⫺4 3.2 ⫻ 10⫺3 1.0 ⫻ 100 8.3 ⫻ 102 Threshold of hearing Whisper Normal conversation Heavy traffic Jackhammer Threshold of pain Jet plane with afterburner Sound Intensity Find the number of decibels from a whisper with sound intensity 5.20 ⫻ 10⫺10 watt per square meter. Compute the answer to two decimal places. We use the decibel formula (1): D ⫽ 10 log ⫽ 10 log I I0 5.2 ⫻ 10⫺10 10⫺12 ⫽ 10 log 520 ⫽ 27.16 decibels MATCHED PROBLEM 4 Find the number of decibels from a jackhammer with sound intensity 3.2 ⫻ 10⫺3 watt per square meter. Compute the answer to two decimal places. 4-6 Common and Natural Logarithms Explore/Discuss 4 309 Imagine using a large sheet of graph paper, ruled with horizontal and vertical lines 81 inch apart, to plot the sound intensities of Table 1 on the x axis and the corresponding decibel levels on the y axis. Suppose that each 81-inch unit on the x axis represents the intensity of the least audible sound (10⫺12 W/m2), and each 18-inch unit on the y axis represents 1 decibel. If the point corresponding to a jet plane with afterburner is plotted on the graph paper, how far is it from the x axis? From the y axis? (Give the first answer in inches and the second in miles!) Discuss. Earthquake Intensity The energy released by the largest earthquake recorded, measured in joules, is about 100 billion (100,000,000,000) times the energy released by a small earthquake that is barely felt. Over the past 150 years several people from various countries have devised different types of measures of earthquake magnitudes so that their severity could be easily compared. In 1935 the California seismologist Charles Richter devised a logarithmic scale that bears his name and is still widely used in the United States. The magnitude M on the Richter scale* is given as follows: Mⴝ 2 E log 3 E0 Richter scale (2) where E is the energy released by the earthquake, measured in joules, and E0 is the energy released by a very small reference earthquake which has been standardized to be E0 ⫽ 104.40 joules The destructive power of earthquakes relative to magnitudes on the Richter scale is indicated in Table 2. T A B L E 2 The Richter Scale Magnitude on Richter Scale Destructive Power 4.5 ⬍ M ⬍ 4.5 4.5 ⬍ M ⬍ 5.5 5.5 ⬍ M ⬍ 6.5 6.5 ⬍ M ⬍ 7.5 7.5 ⬍ M Small Moderate Large Major Greatest *Originally, Richter defined the magnitude of an earthquake in terms of logarithms of the maximum seismic wave amplitude, in thousandths of a millimeter, measured on a standard seismograph. Formula (2) gives essentially the same magnitude that Richter obtained for a given earthquake but in terms of logarithms of the energy released by the earthquake. 310 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS EXAMPLE Earthquake Intensity 5 Solution The 1906 San Francisco earthquake released approximately 5.96 ⫻ 1016 joules of energy. What was its magnitude on the Richter scale? Compute the answer to two decimal places. We use the magnitude formula (2): M⫽ ⫽ 2 E log 3 E0 2 5.96 ⫻ 1016 log 3 104.40 ⫽ 8.25 MATCHED PROBLEM 5 The 1985 earthquake in central Chile released approximately 1.26 ⫻ 1016 joules of energy. What was its magnitude on the Richter scale? Compute the answer to two decimal places. EXAMPLE Earthquake Intensity 6 Solution If the energy release of one earthquake is 1,000 times that of another, how much larger is the Richter scale reading of the larger than the smaller? Let M1 ⫽ 2 E1 log 3 E0 M2 ⫽ and 2 E2 log 3 E0 be the Richter equations for the smaller and larger earthquakes, respectively. Substituting E2 ⫽ 1,000E1 into the second equation, we obtain M2 ⫽ 2 1,000E1 log 3 E0 冢 ⫽ 2 E1 log 103 ⫹ log 3 E0 ⫽ 2 2 E1 (3) ⫹ log 3 3 E0 冣 ⫽ 2 ⫹ M1 Thus, an earthquake with 1,000 times the energy of another has a Richter scale reading of 2 more than the other. MATCHED PROBLEM 6 If the energy release of one earthquake is 10,000 times that of another, how much larger is the Richter scale reading of the larger than the smaller? 4-6 Common and Natural Logarithms 311 Rocket Flight Theory The theory of rocket flight uses advanced mathematics and physics to show that the velocity v of a rocket at burnout (depletion of fuel supply) is given by v ⴝ c ln Wt Wb Rocket equation (3) where c is the exhaust velocity of the rocket engine, Wt is the takeoff weight (fuel, structure, and payload), and Wb is the burnout weight (structure and payload). Because of the Earth’s atmospheric resistance, a launch vehicle velocity of at least 9.0 kilometers per second is required to achieve the minimum altitude needed for a stable orbit. It is clear that to increase velocity v, either the weight ratio Wt /Wb must be increased or the exhaust velocity c must be increased. The weight ratio can be increased by the use of solid fuels, and the exhaust velocity can be increased by improving the fuels, solid or liquid. EXAMPLE 7 Solution Rocket Flight Theory A typical single-stage, solid-fuel rocket may have a weight ratio Wt/Wb ⫽ 18.7 and an exhaust velocity c ⫽ 2.38 kilometers per second. Would this rocket reach a launch velocity of 9.0 kilometers per second? We use the rocket equation (3): v ⫽ c ln Wt Wb ⫽ 2.38 ln 18.7 ⫽ 6.97 kilometers per second The velocity of the launch vehicle is far short of the 9.0 kilometers per second required to achieve orbit. This is why multiple-stage launchers are used—the deadweight from a preceding stage can be jettisoned into the ocean when the next stage takes over. MATCHED PROBLEM 7 A launch vehicle using liquid fuel, such as a mixture of liquid hydrogen and liquid oxygen, can produce an exhaust velocity of c ⫽ 4.7 kilometers per second. However, the weight ratio Wt /Wb must be low—around 5.5 for some vehicles— because of the increased structural weight to accommodate the liquid fuel. How much more or less than the 9.0 kilometers per second required to reach orbit will be achieved by this vehicle? Answers to Matched Problems 1. (A) ⫺1.868 734 (B) 3.356 663 (C) Not possible 2. (A) 14.275 (B) 1.022 3. (A) x ⫽ 0.006 333 (B) x ⫽ 1.21 ⫻ 1012 4. 95.05 decibels 5. 7.80 6. 2.67 (C) 1.022 7. 1 km/s less 312 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS EXERCISE 4-6 C A In Problems 37–40, find domain and range, x and y intercepts, and asymptotes. Round all approximate values to two decimal places. In Problems 1–8, evaluate to four decimal places. 1. log 82,734 2. log 843,250 37. f(x) ⫽ ⫺2 ⫹ ln (1 ⫹ x2) 38. f(x) ⫽ 2 ⫺ ln (1 ⫹ ⱍxⱍ) 39. f(x) ⫽ 1 ⫹ ln (1 ⫺ x ) 40. f(x) ⫽ ⫺1 ⫹ ln ( 1 ⫺ x2 ) ⱍ 2 3. log 0.001 439 4. log 0.035 604 5. ln 43.046 6. ln 2,843,100 7. ln 0.081 043 8. ln 0.000 032 4 In Problems 9–16, evaluate x to four significant digits, given: 9. log x ⫽ 5.3027 10. log x ⫽ 1.9168 ⱍ 41. Find the fallacy. 1⬍3 1 27 3 ⬍ 27 1 27 ⬍ 19 Divide both sides by 27. (13)3 ⬍ (13)2 11. log x ⫽ ⫺3.1773 12. log x ⫽ ⫺2.0411 log (13)3 ⬍ log (13)2 13. ln x ⫽ 3.8655 14. ln x ⫽ 5.0884 3 log 13 ⬍ 2 log 13 15. ln x ⫽ ⫺0.3916 16. ln x ⫽ ⫺4.1083 3⬍2 Divide both sides by log 13. 42. Find the fallacy. B 3⬎2 In Problems 17–24, evaluate to three decimal places. 17. n ⫽ log 2 log 1.15 18. n ⫽ log 2 log 1.12 19. n ⫽ ln 3 ln 1.15 20. n ⫽ ln 4 ln 1.2 ln 0.1 22. x ⫽ ⫺0.0025 ln 0.5 21. x ⫽ ⫺0.21 23. t ⫽ ln 150 ln 3 24. t ⫽ log 200 log 2 3 log 12 ⬎ 2 log 12 Multiply both sides by log 12. log (12)3 ⬎ log (12)2 (12)3 ⬎ (12)2 1 8 ⬎ 14 1⬎2 Multiply both sides by 8. 43. The function f(x) ⫽ log x increases extremely slowly as x → ⬁, but the composite function g(x) ⫽ log (log x) increases still more slowly. (A) Illustrate this fact by computing the values of both functions for several large values of x. In Problems 25–32, evaluate x to five significant digits. (B) Determine the domain and range of the function g. 25. x ⫽ log (5.3147 ⫻ 10 ) (C) Discuss the graphs of both functions. 12 44. The function f(x) ⫽ ln x increases extremely slowly as x → ⬁, but the composite function g(x) ⫽ ln (ln x) increases still more slowly. 26. x ⫽ log (2.0991 ⫻ 1017) 27. x ⫽ ln (6.7917 ⫻ 10⫺12) 28. x ⫽ ln (4.0304 ⫻ 10⫺8) 29. log x ⫽ 32.068 523 30. log x ⫽ ⫺12.731 64 31. ln x ⫽ ⫺14.667 13 32. ln x ⫽ 18.891 143 (A) Illustrate this fact by computing the values of both functions for several large values of x. (B) Determine the domain and range of the function g. (C) Discuss the graphs of both functions. In Problems 33–36, find f ⫺1. Check by graphing f, f ⫺1, and y ⫽ x in the same viewing window on a graphing utility. In Problems 45–48, use a graphing utility to find the coordinates of all points of intersection to two decimal places. 33. f(x) ⫽ 2 ln (x ⫹ 2) 34. f(x) ⫽ 2 ln x ⫹ 2 45. f(x) ⫽ ln x, g(x) ⫽ 0.1x ⫺ 0.2 35. f(x) ⫽ 4 ln x ⫺ 3 36. f(x) ⫽ 4 ln (x ⫺ 3) 46. f(x) ⫽ log x, g(x) ⫽ 4 ⫺ x2 4-6 Common and Natural Logarithms 47. f(x) ⫽ ln x, g(x) ⫽ x1/3 ing of 8.3. How many times more powerful was the Anchorage earthquake than the Long Beach earthquake? 48. f(x) ⫽ 3 ln (x ⫺2), g(x) ⫽ 4e⫺x ★★ The polynomials in Problems 49–52, called Taylor polynomials, can be used to approximate the function g(x) ⫽ ln (1 ⫹ x). To illustrate this approximation graphically, in each problem, graph g(x) ⫽ ln (1 ⫹ x) and the indicated polynomial in the same viewing window, ⫺1 ⱕ x ⱕ 3 and ⫺2 ⱕ y ⱕ 2. 50. P2(x) ⫽ x ⫺ 12 x2 ⫹ 13 x3 62. Space Vehicles. A liquid-fuel rocket has a weight ratio Wt /Wb ⫽ 6.2 and an exhaust velocity c ⫽ 5.2 kilometers per second. What is its velocity at burnout? Compute the answer to two decimal places. 51. P3(x) ⫽ x ⫺ 12 x2 ⫹ 13 x3 ⫺ 14 x4 52. P4(x) ⫽ x ⫺ 12 x2 ⫹ 13 x3 ⫺ 14 x4 ⫹ 15 x5 63. Chemistry. The hydrogen ion concentration of a substance is related to its acidity and basicity. Because hydrogen ion concentrations vary over a very wide range, logarithms are used to create a compressed pH scale, which is defined as follows: APPLICATIONS 53. Sound. What is the decibel level of (A) The threshold of hearing, 1.0 ⫻ 10⫺12 watt per square meter? pH ⫽ ⫺log [H⫹] (B) The threshold of pain, 1.0 watt per square meter? where [H⫹] is the hydrogen ion concentration, in moles per liter. Pure water has a pH of 7, which means it is neutral. Substances with a pH less than 7 are acidic, and those with a pH greater than 7 are basic. Compute the pH of each substance listed, given the indicated hydrogen ion concentration. Compute answers to two significant digits. 54. Sound. What is the decibel level of (A) A normal conversation, 3.2 ⫻ 10⫺6 watt per square meter? (B) A jet plane with an afterburner, 8.3 ⫻ 102 watts per square meter? (A) Seawater, 4.63 ⫻ 10⫺9 (B) Vinegar, 9.32 ⫻ 10⫺4 Compute answers to two significant digits. Also, indicate whether it is acidic or basic. Compute answers to one decimal place. 55. Sound. If the intensity of a sound from one source is 1,000 times that of another, how much more is the decibel level of the louder sound than the quieter one? 64. Chemistry. Refer to Problem 63. Compute the pH of each substance below, given the indicated hydrogen ion concentration. Also, indicate whether it is acidic or basic. Compute answers to one decimal place. 56. Sound. If the intensity of a sound from one source is 10,000 times that of another, how much more is the decibel level of the louder sound than the quieter one? (A) Milk, 2.83 ⫻ 10⫺7 (B) Garden mulch, 3.78 ⫻ 10⫺6 ★ 65. Ecology. Refer to Problem 63. Many lakes in Canada and the United States will no longer sustain some forms of wildlife because of the increase in acidity of the water from acid rain and snow caused by sulfur dioxide emissions from industry. If the pH of a sample of rainwater is 5.2, what is its hydrogen ion concentration in moles per liter? Compute the answer to two significant digits. ★ 66. Ecology. Refer to Problem 63. If normal rainwater has a pH of 5.7, what is its hydrogen ion concentration in moles per liter? Compute the answer to two significant digits. 58. Earthquakes. Anchorage, Alaska, had a major earthquake in 1964 that released 7.08 ⫻ 1016 joules of energy. What was its magnitude on the Richter scale? Compute the answer to one decimal place. ★★ 59. Earthquakes. The 1933 Long Beach, California, earthquake had a Richter scale reading of 6.3, and the 1964 Anchorage, Alaska, earthquake had a Richter scale read- 60. Earthquakes. Generally, an earthquake requires a magnitude of over 5.6 on the Richter scale to inflict serious damage. How many times more powerful than this was the great 1906 Colombia earthquake, which registered a magnitude of 8.6 on the Richter scale? 61. Space Vehicles. A new solid-fuel rocket has a weight ratio Wt /Wb ⫽ 19.8 and an exhaust velocity c ⫽ 2.57 kilometers per second. What is its velocity at burnout? Compute the answer to two decimal places. 49. P1(x) ⫽ x ⫺ 12 x2 57. Earthquakes. The largest recorded earthquake to date was in Colombia in 1906, with an energy release of 1.99 ⫻ 1017 joules. What was its magnitude on the Richter scale? Compute the answer to one decimal place. 313

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