Question

evaluate the logarithmic expression. \\(\log_{8} \frac{1}{64} = \square\\) (type an integer or a simplified fraction.)

Explanation:

Step1: Recall logarithm property

Recall that \(\log_{a}b = c\) is equivalent to \(a^{c}=b\). Also, recall the property \(\log_{a}\frac{1}{x}=-\log_{a}x\). So, \(\log_{8}\frac{1}{64}=-\log_{8}64\).

Step2: Express 64 as a power of 8

We know that \(8^{2}=64\), so \(\log_{8}64 = \log_{8}8^{2}\).

Step3: Apply logarithm power rule

Using the power rule of logarithms \(\log_{a}a^{x}=x\), we have \(\log_{8}8^{2}=2\).

Step4: Find the value of the original expression

Since \(\log_{8}\frac{1}{64}=-\log_{8}64\) and \(\log_{8}64 = 2\), then \(\log_{8}\frac{1}{64}=- 2\).

Answer:

\(-2\)