Solve the logarithmic equation.
\[
x=\log _{4} \frac{1}{64}
\]

Step 1 ：The logarithmic equation is given as \(x = \log_4 \frac{1}{64}\).

Step 2 ：This equation is in the form of \(x = \log_b a\), where \(b = 4\), \(a = \frac{1}{64}\), and \(x\) is what we're trying to solve for.

Step 3 ：The logarithm \(\log_b a = x\) is equivalent to the exponential equation \(b^x = a\). So, we can rewrite the given equation as \(4^x = \frac{1}{64}\) and solve for \(x\).

Step 4 ：By solving the equation, we find that \(x = -3\). This means that \(4^{-3} = \frac{1}{64}\), which is true.

Step 5 ：Therefore, the solution to the logarithmic equation \(x = \log_4 \frac{1}{64}\) is \(x = -3\).

Step 6 ：Final Answer: \(\boxed{-3}\)