jmc

First, we simplify the left side, and we have $$2^x+2^x+2^x+2^x=4\cdot 2^x=2^2\cdot 2^x=2^{x+2}.$$Noting that $512=2^9$, our equation now is $2^{x+2}=2^9$, so $x+2=9$. Therefore, $x=\boxed{7}$.