1Cademy - Try It 10.41: Evaluating $$\log_{12} 144$$, $$\log_4 2$$, and $$\log

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Evaluating a Logarithm by Converting to Exponential Form

Example

Try It 10.41: Evaluating $\log_{12} 144$ , $\log_4 2$ , and $\log_2 \frac{1}{32}$

To evaluate the logarithms $\log_{12} 144$ , $\log_4 2$ , and $\log_2 \frac{1}{32}$ without a calculator, set each expression equal to a variable $x$ and convert it to its exponential form. For $\log_{12} 144 = x$ , the exponential form is $12^x = 144$ . Since $144 = 12^2$ , it follows that $x = 2$ . For $\log_4 2 = x$ , the exponential form is $4^x = 2$ . Since $4^{\frac{1}{2}} = 2$ , $x = \frac{1}{2}$ . For $\log_2 \frac{1}{32} = x$ , the exponential form is $2^x = \frac{1}{32}$ . Because $\frac{1}{32} = 2^{-5}$ , $x = -5$ .