Proof by contradiction$,-,$functional equation

Show that there does not exist a strictly increasing function $f : N → N $ satisfying $f (2) = 3$ and $f (mn) = f (m)f (n)$ for all $m, n ∈ N$.

My Attempt

It can be found that $f(2^x)=3^x$ from which we can conclude that $f(x)=3^{\log_2 x}$ and by which we can argue that for all natural $x$, $f(x)$ is not always natural, so contradiction hence such a function doesn't exist.
Is this proof correct? If not what should be changed?