Answer
$\frac{1}{81}$.
Work Step by Step
${{\log }_{3}}x=-4$.
Simplify the logarithm ${{\log }_{3}}x=-4$ as follows.
${{\log }_{3}}x=-4$
Use the fact that the expression ${{\log }_{a}}x=m\text{ is equivalent to }{{a}^{m}}=x$. Therefore,
$\begin{align}
& x={{3}^{-4}} \\
& =\frac{1}{{{3}^{4}}} \\
& =\frac{1}{81}
\end{align}$
Check:
Substitute $x=\frac{1}{81}$ in the given equation.
$\begin{matrix}
{{\log }_{3}}\left( \frac{1}{81} \right)\overset{?}{\mathop{=}}\,-4 \\
\text{ }\left( \frac{1}{81} \right)\overset{?}{\mathop{=}}\,{{\left( 3 \right)}^{-4}} \\
\text{ }\left( \frac{1}{81} \right)=\frac{1}{81} \\
\end{matrix}$
Thus, the obtained solution is correct.
