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?
If $f(3) < 6$, then $27 = f(8) < f(9)\le 25$, a contradiction. Now, assume that $f(3) > 6$. Then $343 = 7^3 \le f(3)^3 = f(27) < f(32) = 3^5 = 243$. Again a contradiction. Thus, $f(3) = 6$. But then $7776 = 6^5 = f(3)^5 = f(243) < f(256) = f(2^8) = 3^8 = 6561$. Done.
Most of the work was done by quasi and Daniel Fischer (see the comments).