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### Section 6.2 : Logarithm Functions

9. Without using a calculator determine the exact value of $${\log _2} \displaystyle \frac{1}{8}$$.

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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,

${\log _2}\frac{1}{8} = ?\hspace{0.25in} \Rightarrow \hspace{0.25in}{2^?} = \frac{1}{8}$ Show Step 2

Now, we know that if we raise an integer to a negative exponent we’ll get a fraction and so we must have a negative exponent and then we know that $${2^3} = 8$$. Therefore, we can see that $${2^{ - 3}} = \frac{1}{8}$$ and so we must have,

$\require{bbox} \bbox[2pt,border:1px solid black]{{{{\log }_2}\frac{1}{8} = - 3}}$