jmc

Rewrite $3^{2x}$ as $(3^2{)}^x=9^x$, and subtract $9^x$ from both sides to obtain $19=10^x-9^x$. There are no solutions of this equation for $x\le 0$, because neither $10^x$ nor $9^x$ is greater than 1 if $x\le 0$. Trying $x=1$, $x=2$, and $x=3$, we see that $10^x-9^x$ is increasing for $x>0$, and it equals 19 when $x=\boxed{2}$.

Note: Using calculus, we could prove that $10^x-9^x$ is monotonically increasing for $x>0$, which would prove that the solution we found is unique.