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Section 1.8 : Logarithm Functions

3. Without using a calculator determine the exact value of \(\displaystyle {\log _2}\frac{1}{8}\).

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,

\[{\log _2}\frac{1}{8} = ?\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}{2^?} = \frac{1}{8}\]

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,

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