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Section 1.8 : Logarithm Functions

4. Without using a calculator determine the exact value of \({\log _{\frac{1}{4}}}16\).

Hint : Recall that converting a logarithm to exponential form can often help to evaluate these kinds of logarithms.
Show Solution

Converting the logarithm to exponential form gives,

\[{\log _{\frac{1}{4}}}16 = ?\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}{\left( {\frac{1}{4}} \right)^?} = 16\]

Now, we know that if we raise an fraction to a power and get an integer out we must have had a negative exponent. Now, we also know that \({4^2} = 16\). Therefore we can see that \({\left( {\frac{1}{4}} \right)^{ - 2}} = {\left( {\frac{4}{1}} \right)^2} = 16\) and so we must have,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{{{\log }_{\frac{1}{4}}}16 = - 2}}\]