4 EXAMPLE 3 416 Chapter 3 Exponential and Logarithmic Functions Figure 318

4 example 3 416 chapter 3 exponential and logarithmic

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4 EXAMPLE 3 416 Chapter 3 Exponential and Logarithmic Functions Figure 3.18 shows the graphs of and in by viewing rectangles. Are and the same? The graphs illustrate that and have different domains. The graphs are only the same if Thus, we should write ln x 2 = 2 ln x for x 7 0. x 7 0. y = 2 ln x y = ln x 2 2 ln x ln x 2 3 - 5, 5, 1 4 3 - 5, 5, 1 4 y = 2 ln x y = ln x 2 Properties for Expanding Logarithmic Expressions For and 1. Product rule 2. Quotient rule 3. Power rule log b M p = p log b M log b a M N b = log b M - log b N log b 1 MN 2 = log b M + log b N N 7 0: M 7 0 Expand logarithmic expressions.
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Section 3.3 Properties of Logarithms 417 Try to avoid the following errors: Incorrect! log b 1 MN p 2 = p log b 1 MN 2 log b M log b N = log b M - log b N log b a M N b = log b M log b N log b 1 M # N 2 = log b M # log b N log b 1 M - N 2 = log b M - log b N log b 1 M + N 2 = log b M + log b N Study Tip The graphs show that In general, y 2 = ln x + ln 3 y 1 = ln ( x + 3) [ 4, 5, 1] by [ 3, 3, 1] log b 1 M + N 2 Z log b M + log b N . ln (x+3) ln x+ ln 3. y = ln x shifted 3 units left y = ln x shifted ln 3 units up Expanding Logarithmic Expressions Use logarithmic properties to expand each expression as much as possible: a. b. Solution We will have to use two or more of the properties for expanding logarithms in each part of this example. a. Use exponential notation. Use the product rule. Use the power rule. b. Use exponential notation. Use the quotient rule. Use the product rule on Use the power rule. Apply the distributive property. because 2 is the power to which we must raise 6 to get 36. Check Point 4 Use logarithmic properties to expand each expression as much as possible: a. b. log 5 ¢ 1 x 25 y 3 . log b A x 4 1 3 y B 1 6 2 = 36 2 log 6 36 = 2 = 1 3 log 6 x - 2 - 4 log 6 y = 1 3 log 6 x - log 6 36 - 4 log 6 y = 1 3 log 6 x - 1 log 6 36 + 4 log 6 y 2 log 6 1 36 y 4 2 . = log 6 x 1 3 - 1 log 6 36 + log 6 y 4 2 = log 6 x 1 3 - log 6 1 36 y 4 2 log 6 ¢ 1 3 x 36 y 4 = log 6 ¢ x 1 3 36 y 4 = 2 log b x + 1 2 log b y = log b x 2 + log b y 1 2 log b 1 x 2 1 y 2 = log b A x 2 y 1 2 B log 6 ¢ 1 3 x 36 y 4 . log b 1 x 2 1 y 2 EXAMPLE 4
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Condensing Logarithmic Expressions To condense a logarithmic expression , we write the sum or difference of two or more logarithmic expressions as a single logarithmic expression. We use the properties of logarithms to do so. Study Tip These properties are the same as those in the box on page 416. The only difference is that we’ve reversed the sides in each property from the previous box. Properties for Condensing Logarithmic Expressions For and 1. Product rule 2. Quotient rule 3. Power rule p log b M = log b M p log b M - log b N = log b a M N b log b M + log b N = log b 1 MN 2 N 7 0: M 7 0 Condensing Logarithmic Expressions Write as a single logarithm: a. b. Solution a. Use the product rule. We now have a single logarithm.
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  • Fall '16
  • Kirkpatrick
  • Physics, Derivative, pH, Natural logarithm, Logarithm, natural logarithms

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