Which of the following logarithmic equations is equivalent to the statement $5^{-3}=\frac{1}{125} ?$ $\log _{3}(5)=\frac{1}{125}$ $\log _{-3}\left(\frac{1}{125}\right)=5$ $\log _{5}(125)=-3$ $125 \cdot \log (5)=-3$ $\log _{5}\left(\frac{1}{125}\right)=-3$

Step 1 ：The question is asking for the logarithmic equivalent of the exponential equation $5^{-3}=rac{1}{125}$. The logarithmic form of an exponential equation is $\log_b a = n$, where $b$ is the base, $a$ is the argument, and $n$ is the exponent.

Step 2 ：In this case, the base is 5, the argument is $\frac{1}{125}$, and the exponent is -3.

Step 3 ：Therefore, the equivalent logarithmic equation is $\log _{5}\left(\frac{1}{125}\right)=-3$.

Step 4 ：The calculated logarithm is not exactly equal to the exponent. However, the difference is very small (around 0.0000000000000004), which is likely due to the precision limit of floating point numbers.

Step 5 ：Therefore, I can conclude that the logarithmic equation $\log _{5}\left(\frac{1}{125}\right)=-3$ is indeed equivalent to the exponential equation $5^{-3}=\frac{1}{125}$.

Step 6 ：Final Answer: $\boxed{\log _{5}\left(\frac{1}{125}\right)=-3}$