Answer
$- \frac {3}{2}$
Work Step by Step
$\log_5 \frac{1}{\sqrt {125}}$
Rewrite the square root to exponential form
$\log_5 \frac{1}{125^{\frac{1}{2}}}$
125 = $5^{3}$ since 5 $\times$ 5 $\times$ 5 = 125.
$\log_5 \frac{1}{(5)^{{3}^{\frac{1}{2}}}}$
When there's a base to a power times a power, we multiply the powers together
$\log_5 \frac{1}{5^{\frac{3}{2}}}$
We can move the denominator to the numerator by changing the power to a negative
$\log_5 5^{- \frac{3}{2}}$
= $- \frac {3}{2}$