Section 4.3 Properties of Logarithms 437 34. 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 = Critical Thinking Eercises Make Sense? In Eercises 35 38, determine whether each statement makes sense or does not make sense, and eplain your reasoning. 35. I ve noticed that eponential functions and logarithmic functions ehibit inverse, or opposite, behavior in many ways. For eample, a vertical translation shifts an eponential function s horizontal asymptote and a horizontal translation shifts a logarithmic function s vertical asymptote. 36. I estimate that log lies between and because 8 8 6 = 8 and 8 = 64. 37. I can evaluate some common logarithms without having to use a calculator. 38. An earthquake of magnitude 8 on the Richter scale is twice as intense as an earthquake of magnitude 4. In Eercises 39 4, 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 39. log 4 = 8 4 40. log-00 = - 4. The domain of f = log is - q, q. 4. log b is the eponent to which b must be raised to obtain. 43. Without using a calculator, find the eact value of log 3 8 - log p log 8 - log 0.00. 44. Without using a calculator, find the eact value of log 4 3log 3 log 84. 45. Without using a calculator, determine which is the greater number: log 4 60 or log 3 40. Group Eercise 46. This group eercise involves eploring the way we grow. Group members should create a graph for the function that models the percentage of adult height attained by a boy who is years old, f = 9 + 48.8 log +. Let =,, 3, Á,, find function values, and connect the resulting points with a smooth curve. Then create a graph for the function that models the percentage of adult height attained by a girl who is years old, g = 6 + 35 log - 4. Let = 5, 6, 7, Á, 5, find function values, and connect the resulting points with a smooth curve. Group members should then discuss similarities and differences in the growth patterns for boys and girls based on the graphs. Preview Eercises Eercises 47 49 will help you prepare for the material covered in the net section. In each eercise, evaluate the indicated logarithmic epressions without using a calculator. 47. a. Evaluate: log 3. b. Evaluate: log 8 + log 4. c. What can you conclude about log 3, or log 8 # 4? 48. a. Evaluate: log 6. b. Evaluate: log 3 - log. c. What can you conclude about log 6, or log a 3 b? 49. a. Evaluate: log 3 8. b. Evaluate: log 3 9. c. What can you conclude about log 3 8, or log 3 9? Section 4.3 Objectives Use the product rule. Use the quotient rule. Use the power rule. Epand logarithmic epressions. Condense logarithmic epressions. Use the change-of-base property. Properties of Logarithms We all learn new things in different ways. In this section, we consider important properties of logarithms. What would be the most effective way for you to learn these properties? Would it be helpful to use your graphing utility and discover one of these properties for yourself? To do so, work Eercise 33 in Eercise Set 4. before continuing. Would it be helpful to evaluate certain logarithmic epressions that suggest three of the properties? If this is the case, work Preview Eercises 47 49 in Eercise Set 4. before continuing. Would the properties become more meaningful if you could see eactly where they come from? If so, you will find details of the proofs of many of these properties in the appendi. The remainder of our work in this chapter will be based on the properties of logarithms that you learn in this section.
438 Chapter 4 Eponential and Logarithmic Functions Use the product rule. Discovery We know that log 00,000 = 5. Show that you get the same result by writing 00,000 as 000 # 00 and then using the product rule. Then verify the product rule by using other numbers whose logarithms are easy to find. The Product Rule Properties of eponents correspond to properties of logarithms. For eample, when we multiply with the same base, we add eponents: b m # b n = b m + n. This property of eponents, coupled with an awareness that a logarithm is an eponent, suggests the following property, called the product rule: The Product Rule Let b, M, and N be positive real numbers with b Z. log b MN = log b M + log b N The logarithm of a product is the sum of the logarithms. When we use the product rule to write a single logarithm as the sum of two logarithms, we say that we are epanding a logarithmic epression. For eample, we can use the product rule to epand ln7: ln (7) = ln 7 + ln. The logarithm of a product is the sum of the logarithms. EXAMPLE Using the Product Rule Use the product rule to epand each logarithmic epression: a. log 4 7 # 5 b. log0. a. log 4 7 # 5 = log4 7 + log 4 5 The logarithm of a product is the sum of the logarithms. b. log0 = log 0 + log = + log The logarithm of a product is the sum of the logarithms. These are common logarithms with base 0 understood. Because log b b =, then log 0 =. Check Point Use the product rule to epand each logarithmic epression: a. log 6 7 # b. log00. Use the quotient rule. Discovery We know that log 6 = 4. Show that you get the same result by writing 6 3 as and then using the quotient rule. Then verify the quotient rule using other numbers whose logarithms are easy to find. The Quotient Rule When we divide with the same base, we subtract eponents: b m b n = bm - n. This property suggests the following property of logarithms, called the quotient rule: The Quotient Rule Let b, M, and N be positive real numbers with b Z. log b a M N b = log b M - log b N The logarithm of a quotient is the difference of the logarithms.
Section 4.3 Properties of Logarithms 439 When we use the quotient rule to write a single logarithm as the difference of two logarithms, we say that we are epanding a logarithmic epression. For eample, we can use the quotient rule to epand log : log a b = log - log. The logarithm of a quotient is the difference of the logarithms. EXAMPLE Using the Quotient Rule Use the quotient rule to epand each logarithmic epression: a. log 7a 9 b. b ln e3 7. a. b. log 7 a 9 b = log 7 9 - log 7 ln e3 7 = ln e3 - ln 7 = 3 - ln 7 The logarithm of a quotient is the difference of the logarithms. The logarithm of a quotient is the difference of the logarithms. These are natural logarithms with base e understood. Because ln e =, then ln e 3 = 3. Check Point Use the quotient rule to epand each logarithmic epression: a. log 8a 3 b. b ln e5. Use the power rule. The Power Rule When an eponential epression is raised to a power, we multiply eponents: b m n = b mn. This property suggests the following property of logarithms, called the power rule: The Power Rule Let b and M be positive real numbers with b Z, and let p be any real number. log b M p = p log b M The logarithm of a number with an eponent is the product of the eponent and the logarithm of that number. When we use the power rule to pull the eponent to the front, we say that we are epanding a logarithmic epression. For eample, we can use the power rule to epand ln : ln = ln. The logarithm of a number with an eponent is the product of the eponent and the logarithm of that number.
440 Chapter 4 Eponential and Logarithmic Functions Figure 4.8 shows the graphs of y = ln and y = ln in 3-5, 5, 4 by 3-5, 5, 4 viewing rectangles. Are ln and ln the same? The graphs illustrate that y = ln and y = ln have different domains. The graphs are only the same if 7 0. Thus, we should write ln = ln for 7 0. y = ln y = ln Domain: (, 0) (0, ) Figure 4.8 ln and ln have different domains. Domain: (0, ) When epanding a logarithmic epression, you might want to determine whether the rewriting has changed the domain of the epression. For the rest of this section, assume that all variables and variable epressions represent positive numbers. EXAMPLE 3 Using the Power Rule Use the power rule to epand each logarithmic epression: a. log 5 7 4 b. ln c. a. log 5 7 4 = 4 log 5 7 The logarithm of a number with an eponent is the eponent times the logarithm of the number. b. ln = ln Rewrite the radical using a rational eponent. = ln log4 5. Use the power rule to bring the eponent to the front. c. log4 5 = 5 log4 We immediately apply the power rule because the entire variable epression, 4, is raised to the 5th power. Check Point 3 Use the power rule to epand each logarithmic epression: a. log 6 3 9 b. ln3 c. log + 4. Epand logarithmic epressions. Epanding Logarithmic Epressions It is sometimes necessary to use more than one property of logarithms when you epand a logarithmic epression. Properties for epanding logarithmic epressions are as follows: Properties for Epanding Logarithmic Epressions For M 7 0 and N 7 0:. log b MN = log b M + log b N Product rule. log b a M Quotient rule N b = log b M - log b N 3. log b M p = p log b M Power rule
Section 4.3 Properties of Logarithms 44 Study Tip The graphs show that In general, ln (+3) ln +ln 3. y = ln shifted 3 units left log b M + N Z log b M + log b N. y = ln ( + 3) y = ln shifted ln 3 units up y = ln + ln 3 Try to avoid the following errors: Incorrect! log b M + N = log b M + log b N log b M - N = log b M - log b N log b M # N = logb M # logb N log b a M N b = log b M log b N log b M log b N = log b M - log b N log b MN p = p log b MN [ 4, 5, ] by [ 3, 3, ] EXAMPLE 4 Epanding Logarithmic Epressions Use logarithmic properties to epand each epression as much as possible: a. log b y b. log 6 3 36y 4. We will have to use two or more of the properties for epanding logarithms in each part of this eample. a. log b y = log b A y B Use eponential notation. = log b + log b y Use the product rule. = log b + log b y Use the power rule. b. log 6 3 36y 4 = log 6 3 36y 4 Use eponential notation. = log 6 3 - log 6 36y 4 = log 6 3 - log 6 36 + log 6 y 4 = 3 log 6 - log 6 36 + 4 log 6 y Use the quotient rule. Use the product rule on log 6 36y 4. Use the power rule. = 3 log 6 - log 6 36-4 log 6 y Apply the distributive property. = 3 log 6 - - 4 log 6 y log 6 36 = because is the power to which we must raise 6 to get 36. 6 = 36 Check Point 4 Use logarithmic properties to epand each epression as much as possible: a. log ba 4 3 yb b. log 5 5y 3.
44 Chapter 4 Eponential and Logarithmic Functions Condense logarithmic epressions. Study Tip These properties are the same as those in the bo on page 440. The only difference is that we ve reversed the sides in each property from the previous bo. Condensing Logarithmic Epressions To condense a logarithmic epression, we write the sum or difference of two or more logarithmic epressions as a single logarithmic epression. We use the properties of logarithms to do so. Properties for Condensing Logarithmic Epressions For M 7 0 and N 7 0:. log b M + log b N = log b MN Product rule. log b M - log b N = log b a M Quotient rule N b 3. p log b M = log b M p Power rule EXAMPLE 5 Condensing Logarithmic Epressions Write as a single logarithm: a. log 4 + log 4 3 b. log4-3 - log. a. log 4 + log 4 3 = log 4 # 3 = log 4 64 = 3 Use the product rule. We now have a single logarithm. However, we can simplify. log because 4 3 4 64 = 3 = 64. b. log4-3 - log = loga 4-3 b Use the quotient rule. Check Point 5 Write as a single logarithm: a. log 5 + log 4 b. log7 + 6 - log. Coefficients of logarithms must be before you can condense them using the product and quotient rules. For eample, to condense ln + ln +, the coefficient of the first term must be. We use the power rule to rewrite the coefficient as an eponent:. Use the power rule to make the number in front an eponent. ln +ln (+)=ln +ln (+)=ln [ (+)].. Use the product rule. The sum of logarithms with coefficients of is the logarithm of the product. EXAMPLE 6 Condensing Logarithmic Epressions Write as a single logarithm: a. log + 4 log - b. c. 4 log b - log b 6 - log b y. 3 ln + 7 - ln
a. b. c. log + 4 log - = log + log - 4 = logc - 4 D 3 ln + 7 - ln = ln + 7 3 - ln + 73 = lnb R 4 log b - log b 6 - log b y = log b 4 - log b 6 - log b y = log b 4 - A logb 36 + log b y B = log b 4 - log b A 36y B Section 4.3 Properties of Logarithms 443 Use the power rule so that all coefficients are. Use the product rule. The condensed form can be epressed as log3-4 4. Use the power rule so that all coefficients are. Use the quotient rule. Use the power rule so that all coefficients are. Rewrite as a single subtraction. Use the product rule. = log b 4 36y 4 or log b 36y Use the quotient rule. Check Point 6 Write as a single logarithm: a. ln + 3 ln + 5 b. log - 3 - log c. 4 log b - log b 5-0 log b y. Use the change-of-base property. The Change-of-Base Property We have seen that calculators give the values of both common logarithms (base 0) and natural logarithms (base e). To find a logarithm with any other base, we can use the following change-of-base property: The Change-of-Base Property For any logarithmic bases a and b, and any positive number M, log b M = log a M log a b. The logarithm of M with base b is equal to the logarithm of M with any new base divided by the logarithm of b with that new base. In the change-of-base property, base b is the base of the original logarithm. Base a is a new base that we introduce. Thus, the change-of-base property allows us to change from base b to any new base a, as long as the newly introduced base is a positive number not equal to. The change-of-base property is used to write a logarithm in terms of quantities that can be evaluated with a calculator. Because calculators contain keys for common (base 0) and natural (base e) logarithms, we will frequently introduce base 0 or base e. Change-of-Base Property log b M= log a M log a b a is the new introduced base. Introducing Common Logarithms log b M= log 0 M log 0 b 0 is the new introduced base. Introducing Natural Logarithms log b M= log e M log e b e is the new introduced base.
444 Chapter 4 Eponential and Logarithmic Functions Using the notations for common logarithms and natural logarithms, we have the following results: The Change-of-Base Property: Introducing Common and Natural Logarithms Introducing Common Logarithms log b M = log M log b Introducing Natural Logarithms log b M = ln M ln b EXAMPLE 7 Changing Base to Common Logarithms Discovery Find a reasonable estimate of log 5 40 to the nearest whole number. To what power can you raise 5 in order to get 40? Compare your estimate to the value obtained in Eample 7. Use common logarithms to evaluate log 5 40. Because log 5 40 = This means that log 5 40 L 3.07. log b M = log M log b, log 40 log 5 L 3.07. Use a calculator: 40 LOG 5 LOG = or LOG 40 LOG 5 ENTER. On some calculators, parentheses are needed after 40 and 5. Check Point 7 Use common logarithms to evaluate log 7 506. EXAMPLE 8 Changing Base to Natural Logarithms Technology Use natural logarithms to evaluate log 5 40. Because log b M = ln M ln b, ln 40 log 5 40 = ln 5 L 3.07. We have again shown that log 5 40 L 3.07. Use a calculator: or LN 40 LN 5 ENTER. On some calculators, parentheses are needed after 40 and 5. Check Point 8 Use natural logarithms to evaluate log 7 506. We can use the change-of-base property to graph logarithmic functions with bases other than 0 or e on a graphing utility. For eample, Figure 4.9 shows the graphs of in a [0, 0, ] by 3-3, 3, 4 viewing rectangle. Because log and log 0 = ln = ln the functions ln ln 0, are entered as and y = log and y = log 0 y = LN LN y = LN LN 0. On some calculators, parentheses are needed after,, and 0. 40 LN 5 LN = y = log 0 y = log Figure 4.9 Using the change-of-base property to graph logarithmic functions
Section 4.3 Properties of Logarithms 445 Eercise Set 4.3 Practice Eercises In Eercises 40, use properties of logarithms to epand each logarithmic epression as much as possible. Where possible, evaluate logarithmic epressions without using a calculator.. log5 7 # 3. log 8 3 # 7 3. 4. log 9 9 5. log 000 6. 7. log 7a 7 8. log 9a 9 9. b b 0. loga. log 4a 64. 000 b y b 3. e ln 5 4. ln e4 8 5. log b 3 6. log b 7 7. log N -6 8. log M -8 9. ln5 0. ln7. log b y. log by 3 3. log 4 4. 64 5. log 36 6. log 64 7. log b y 6 8 z + + 8. log b 3 y 9. log00 30. lne z 3. log 3. log 33. log b y3 A 3 y A 5 y z 3 3 y4 y y 4 34. log b 35. log 5 3 36. log 5 A 5 A 6 37. lnb 3 3 + R 38. + 4 39. logb 0 3 - R 40. 7 + In Eercises 4 70, use properties of logarithms to condense each logarithmic epression. Write the epression as a single logarithm whose coefficient is. Where possible, evaluate logarithmic epressions without using a calculator. 4. log 5 + log 4. log 50 + log 4 43. ln + ln 7 44. ln + ln 3 45. log 96 - log 3 46. log 3 405 - log 3 5 47. log + 5 - log 48. log3 + 7 - log 49. log + 3 log y 50. log + 7 log y 5. ln + ln y 5. 3 ln + ln y 53. log b + 3 log b y 54. 5 log b + 6 log b y 55. 5 ln - ln y 56. 7 ln - 3 ln y 57. 3 ln - 3 ln y 58. ln - ln y 59. 4 ln + 6-3 ln 60. 8 ln + 9-4 ln 6. 3 ln + 5 ln y - 6 ln z 6. 63. 64. 65. 66. 67. 68. 4 ln + 7 ln y - 3 ln z log + log y 3 log 4 - log 4 y log 5 + log 5 y - log 5 + 3 log 4 - log 4 y + log 4 + z 5 3 3 ln + 5 - ln - ln - 44 3 35 ln + 6 - ln - ln - 54 log 7 7 log 0,000 loga 00 b log 5 a 5 y b log 5 5 lnb 4 3 + 3 R + 3 5 logb 003 3 5 - R 3 + 7 69. 70. log + log - - log 7 - log + log + log - 4 - log 5 - log + In Eercises 7 78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. 7. log 5 3 7. log 6 7 73. log 4 87.5 74. log 6 57. 75. log 0. 7 76. log 0.3 9 77. log p 63 78. log p 400 In Eercises 79 8, use a graphing utility and the change-of-base property to graph each function. 79. y = log 3 80. y = log 5 8. y = log + 8. y = log 3 - Practice Plus In Eercises 83 88, let log b = A and log b 3 = C. Write each epression in terms of A and C. 83. 84. log b 6 85. log b 8 log b 3 86. log b 8 87. 3 log b 88. log A 7 b A 6 In Eercises 89 0, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. 89. ln e = 0 90. ln 0 = e 9. log 4 3 = 3 log 4 9. ln8 3 = 3 ln 93. log 0 = 94. ln + = ln + ln 95. ln5 + ln = ln5 96. ln + ln = ln3 97. log + 3 log + 3 - log = log log + 98. 99. 00. 0. = log + - log - log - log 6 - + 4 = log 6 - - log 6 + 4 log 6 34 + 4 = log 6 4 + log 6 + log 0. e = 3 7 = log 7 3 ln Application Eercises 03. The loudness level of a sound can be epressed by comparing the sound s intensity to the intensity of a sound barely audible to the human ear. The formula D = 0log I - log I 0 describes the loudness level of a sound, D, in decibels, where I is the intensity of the sound, in watts per meter, and I 0 is the intensity of a sound barely audible to the human ear. a. Epress the formula so that the epression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question: If a sound has an intensity 00 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound?
446 Chapter 4 Eponential and Logarithmic Functions 04. The formula describes the time, t, in weeks, that it takes to achieve mastery of a portion of a task, where A is the maimum learning possible, N is the portion of the learning that is to be achieved, and c is a constant used to measure an individual s learning style. a. Epress the formula so that the epression in brackets is written as a single logarithm. b. The formula is also used to determine how long it will take chimpanzees and apes to master a task. For eample, a typical chimpanzee learning sign language can master a maimum of 65 signs. Use the form of the formula from part (a) to answer this question: How many weeks will it take a chimpanzee to master 30 signs if c for that chimp is 0.03? Writing in Mathematics 05. Describe the product rule for logarithms and give an eample. 06. Describe the quotient rule for logarithms and give an eample. 07. Describe the power rule for logarithms and give an eample. 08. Without showing the details, eplain how to condense ln - ln +. 09. Describe the change-of-base property and give an eample. 0. Eplain how to use your calculator to find log 4 83.. You overhear a student talking about a property of logarithms in which division becomes subtraction. Eplain what the student means by this.. Find ln using a calculator. Then calculate each of the following: - - + 3 - ; - 4 + + 3 ; - + 3-4 ; 5 ; Á. Describe what you observe. Technology Eercises t = 3ln A - lna - N4 c 3. a. Use a graphing utility (and the change-of-base property) to graph y = log 3. b. Graph y = + log 3, y = log 3 +, and y = -log 3 in the same viewing rectangle as y = log 3. Then describe the change or changes that need to be made to the graph of y = log 3 to obtain each of these three graphs. 4. Graph y = log, y = log0, and y = log0. in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship? 5. Use a graphing utility and the change-of-base property to graph y = log 3, y = log 5, and y = log 00 in the same viewing rectangle. a. Which graph is on the top in the interval (0, )? Which is on the bottom? b. Which graph is on the top in the interval, q? Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using y = log b where b 7. Disprove each statement in Eercises 6 0 by a. letting y equal a positive constant of your choice, and b. using a graphing utility to graph the function on each side of the equal sign. The two functions should have different graphs, showing that the equation is not true in general. 6. 7. log + y = log + log y loga y b = log log y 8. ln - y = ln - ln y 9. lny = ln ln y 0. ln = ln - ln y ln y Critical Thinking Eercises Make Sense? In Eercises 4, determine whether each statement makes sense or does not make sense, and eplain your reasoning.. Because I cannot simplify the epression b m + b n by adding eponents, there is no property for the logarithm of a sum.. Because logarithms are eponents, the product, quotient, and power rules remind me of properties for operations with eponents. 3. I can use any positive number other than in the change-ofbase property, but the only practical bases are 0 and e because my calculator gives logarithms for these two bases. 4. I epanded log 4 by writing the radical using a rational A y eponent and then applying the quotient rule, obtaining log 4 - log 4 y. In Eercises 5 3, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 5. ln = ln log 7 49 6. log 7 7 = log 7 49 - log 7 7 7. log b 3 + y 3 = 3 log b + 3 log b y 8. log b y 5 = log b + log b y 5 9. Use the change-of-base property to prove that log e = ln 0. 30. If log 3 = A and log 7 = B, find log 7 9 in terms of A and B. 3. Write as a single term that does not contain a logarithm: e ln 85 - ln. 3. If f = log b, show that f + h - f = log h b a + h b h, h Z 0. Preview Eercises Eercises 33 35 will help you prepare for the material covered in the net section. 33. Solve for : a - = b + 3. 34. Solve: - 7 = 3. + 35. Solve: 4 + 3 =.