Answer
$x=\frac{4}{9}$
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, $log_{\frac{2}{3}}x=2$ is equivalent to $(\frac{2}{3})^{2}=x$.
Therefore, $x=(\frac{2}{3})^{2}=\frac{2\times2}{3\times3}=\frac{4}{9}$.