Answer
$x=-1$
Work Step by Step
First, we let
$\log_{6}{\frac{1}{6}} = x$
RECALL:
$\log_b{y} =x \longrightarrow b^x=y$
Use the rule above, where $y=\frac{1}{6}$ and $b=6$, to obtain:
$6^x=\frac{1}{6}$
Write $\frac{1}{6}$ as $ 6^{-1}$ to obtain:
$6^x=6^{-1}$
Use the rule "If $a^x=a^y$, then $x=y$" to obtain:
$x=-1$