Evaluate $\log _{\frac{1}{8}}(64)$ without using a calculator.
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Final Answer: $\boxed{-2}$

Steps

Step 1 ：Rewrite the logarithm base $\frac{1}{8}$ of 64 as $\frac{1}{8}^x = 64$

Step 2 ：Recognize that $\frac{1}{8}$ is $2^{-3}$ and 64 is $2^6$

Step 3 ：Substitute these values into the equation to get $(2^{-3})^x = 2^6$

Step 4 ：Simplify this to $2^{-3x} = 2^6$

Step 5 ：Since the bases are the same, equate the exponents to get $-3x = 6$

Step 6 ：Solve for x to get $x = -\frac{6}{3} = -2$

Step 7 ：Final Answer: $\boxed{-2}$