Recall that $\log_b x$ is the "exponent needed on the base $b$ to produce $x$". Here, we are dealing with the natural log, so the base is $e$.

Further, $\sqrt[3]{e^2} = e^{2/3}$.

Hence, the exponent needed on the base $e$ to produce $e^{2/3}$ is of course $\frac{2}{3}$.

Equivalently,

$$\ln \sqrt[3]{e^2} = \frac{2}{3}$$