Answer
$x=\frac{1}{8}$
Work Step by Step
We know that if $b\gt0$ and $b\ne1$, then $y=log_{b}x$ is equivalent to $x=b^{y}$ (where $x\gt0$ and $y$ is a real number).
Therefore, $log_{x}2=-\frac{1}{3}$ is equivalent to $x^{-\frac{1}{3}}=2$.
Therefore, $x=\frac{1}{8}$, because $(\frac{1}{8})^{-\frac{1}{3}}=\frac{1}{(\frac{1}{8})^{\frac{1}{3}}}=\frac{1}{\sqrt[3]\frac{1}{8}}=\frac{1}{\frac{1}{2}}=2$.