Solve the logarithmic equation. \[ \log _{64} \frac{1}{8}=x \]

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 \(x = \boxed{-0.5}\).