We note that $64=2^{6}$ and $512=2^{9}$. Therefore, the equation $512^{x}=64^{240}$ can be rewritten as $(2^{9})^{x}=(2^{6})^{240}$ or $2^{9x}=2^{6(240)}$. Since the bases in this last equation are equal, then the exponents are equal, so $9x=6(240)$ or $x=\frac{1440}{9}=160$.