Evaluate \( \log _{5} \frac{1}{625} \) \( \log _{5} \frac{1}{625}=\square \)

To evaluate \( \log _{5} \frac{1}{625} \), we can rewrite the expression inside the logarithm. Notice that \( 625 = 5^4 \). Therefore, we can express \( \frac{1}{625} \) as: \[ \frac{1}{625} = 625^{-1} = (5^4)^{-1} = 5^{-4} \] Now we can substitute this back into the logarithm: \[ \log_{5} \frac{1}{625} = \log_{5} (5^{-4}) \] Using the power rule of logarithms, which states that \( \log_{b} a^n = n \log_{b} a \), we can simplify this: \[ \log_{5} (5^{-4}) = -4 \log_{5} 5 \] Since \( \log_{5} 5 = 1 \), we have: \[ -4 \log_{5} 5 = -4 \cdot 1 = -4 \] Thus, the final answer is: \[ \log_{5} \frac{1}{625} = -4 \] So, \( \log _{5} \frac{1}{625} = \square -4 \)