Answer
$\frac{2}{3}$
Work Step by Step
The definition of the logarithmic function says that $y=\log_a{x}$ if and only if $a^y=x$. Also, $a\gt0,a\ne1$ and $x\gt0$.
Hence $\log_{5} {\sqrt[3]{25}}=y$, then $\left(5\right)^y=\sqrt[3]{25}$ and we know that $\sqrt[3]{25}=\sqrt[3]{5^2}=\left(5\right)^{\frac{2}{3}}.$
Thus, $\left(5\right)^{y}=\left(5\right)^{\frac{2}{3}}$. We know that $a^b=a^c\longrightarrow b=c$ if $a\ne1,a\ne-1$ (which applies here), hence $y=\frac{2}{3}$.