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### Section 6.2 : Logarithm Functions

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

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

Converting the logarithm to exponential form gives,

${\log _{\frac{1}{4}}}16 = ?\hspace{0.25in} \Rightarrow \hspace{0.25in}{\left( {\frac{1}{4}} \right)^?} = 16$ Show Step 2

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,

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