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 $x\leq 0$, because neither $10^x$ nor $9^x$ is greater than 1 if $x\leq 0$.  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.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Solve for $x: 3^{2x} + 19 = 10^x$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
2
Solve for $x: 3^{2x} + 19 = 10^x$.
test