jmc

In order to find ${\log }_{625}x$, we must first find $x$. We begin by writing ${\log }_9(x-2)=\frac{1}{2}$ in exponential form, which gives us $9^{\frac{1}{2}}=x-2$. Solving for $x$, we find that $x=9^{\frac{1}{2}}+2=3+2=5$. After plugging this value of $x$ into the second expression, the final step is to find ${\log }_{625}5$. Since we know that $625=5^4$ or $625^{\frac{1}{4}}=5$, ${\log }_{625}5=\boxed{\frac{1}{4}}$.