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ss_4_4 - Section 4.4 Properties of Logarithms Properties of...

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Section 4.4. Properties of Logarithms Properties of Logarithms: log a 1 = 0 log a a = 1 log a M r = r log a M log a a r = r a log a x = x log a M = log a N implies M = N log a ( MN ) = log a M + log a N log a ( M N ) = log a M - log a N The following examples illustrate the above rules. Example. log ( x ( x + 1) 2 ) = log x + log ( x + 1) 2 = 1 2 log x + 2 log ( x + 1) Example. log x ( x +1) 2 = log x - log ( x + 1) 2 = 1 2 log x - 2 log ( x + 1) Example. log 3 2 - 3 log 5 = log 3 2 - log 5 3 = log 3 2 5 3 = log 9 125 Example. Suppose y = 3 x 2 , then ln y = ln 3 x 2 = ln x 2 + ln 3 = 2 ln x + ln 3 Interesting observation: In the previous example, y was a power function of x (i.e. y = 3 x 2 ), and this implied that ln y was a linear function of ln x , i.e., y = 3 x 2 implied that ln y = 2 ln x + ln 3. The example illustrates the following general rule: 1
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y = ax b if and only if ln y = b ln x + ln a The above implies that if a plot of ln x against ln y , for some set of data, is a straight line, then one would expect y to be a power function of x for that set of data.
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