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section23_material - MATH 110 BUSINESS CALCULUS CHAPTER 2...

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Unformatted text preview: MATH 110 BUSINESS CALCULUS CHAPTER 2 NONLINEAR MODELS Section 2.3 Logarithmic Functions and Models Exponential Form: Logarithmic Form: Example: by = x log b x = y 43 = 64 can be written as log 4 64 = 3 Common Logarithm and Natural Logarithm Base 10: log10 x = log x Base e: log e x = ln x Change of Base Formula: log b a = log a ln a = log b ln b Example: Approximate log 7 10. Keep four decimal places. log10 log 7 10 = ≈ 1.1833 log 7 Check: 71.1833 = 10.0001 ≈ 10. Other properties of the Logarithms: 1. log b ( xy ) = log b x + log b y 2. ⎛ log b ⎜ ⎝ 3. log b x r = r log b x 4. log b b = 1 5. log b 1 = 0 x⎞ ⎟ = log b x − log b y y⎠ () Example: Find the exact value of the expression: log 6 4 + log 6 9 log 6 4 + log 6 9 = log 6 ( 4 ⋅ 9 ) = log 6 ( 36 ) = 2 Logarithmic Identities: () 1. log b b x = x 2. b logb x = x Example: Rewrite Since 2 = eln 2 Q (t ) = 2t we can write in the form of Q (t ) = e( ln 2 )t Q (t ) = e rt An logarithmic function of variable x is a function that can be written in the form f ( x) = log b x + C where C and b constants ( b ≠ 1, b > 0) We call b the base of the exponential function. Example: The graph of the function f ( x) = log 2 x Compare the logarithmic functions of different bases: ...
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This note was uploaded on 10/19/2011 for the course MATH 110 taught by Professor Staff during the Spring '11 term at S.F. State.

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