Find the exact value of the logarithm without using a calculator. \( \log _{1 / 5} 125 \) \( \log _{1 / 5} 125=\square \) (Type an integer or a simplified fraction.)

To solve \( \log_{1/5} 125 \), we can express 125 as a power of 5: \( 125 = 5^3 \). Then we rewrite the logarithm: \[ \log_{1/5} 125 = \log_{1/5} (5^3) \] Using the property of logarithms that states \( \log_b (a^c) = c \cdot \log_b a \), we have: \[ \log_{1/5} (5^3) = 3 \cdot \log_{1/5} 5 \] Now, we need to find \( \log_{1/5} 5 \). This can be computed by recalling that the base \( \frac{1}{5} \) can be expressed as \( 5^{-1} \): \[ \log_{1/5} 5 = \log_{5^{-1}} 5 \] Since \( \log_{b^{-1}} a = -\log_b a \), we know: \[ \log_{5^{-1}} 5 = -\log_5 5 \] And since \( \log_5 5 = 1 \): \[ \log_{1/5} 5 = -1 \] Putting it all together, we have: \[ \log_{1/5} 125 = 3 \cdot (-1) = -3 \] Thus, the exact value of \( \log_{1/5} 125 \) is \[ \log_{1/5} 125 = -3 \] So, the answer is \( \square = -3 \).