Solve the logarithmic equation.
\[
\log _{64} \frac{1}{8}=x
\]
Answer
Final Answer: The solution to the logarithmic equation \(\log _{64} \frac{1}{8}=x\) is \(x = \boxed{-0.5}\).
Step 1 ：We are given the logarithmic equation \(\log _{64} \frac{1}{8}=x\).
Step 2 ：This equation is in the form of \(\log_b a = x\), which can be rewritten as \(b^x = a\).
Step 3 ：In this case, \(b = 64\), \(a = \frac{1}{8}\), and we need to solve for \(x\).
Step 4 ：Substituting the values of \(b\) and \(a\) into the equation \(b^x = a\), we get \(64^x = \frac{1}{8}\).
Step 5 ：Solving this equation, we find that \(x = -0.5\). This makes sense because \(64^{-0.5} = \frac{1}{8}\).
Step 6 ：Final Answer: The solution to the logarithmic equation \(\log _{64} \frac{1}{8}=x\) is \(x = \boxed{-0.5}\).
