Show that there does not exist a strictly increasing function $f:\mathbb{N}\rightarrow\mathbb{N}$ satisfying
$$f(2)=3$$ $$f(mn)=f(m)f(n)\forall m,n\in\mathbb{N}$$
Progress:
Assume the function exists. Let $f(3)=k$
Since $2^3 < 3^2$,
$$3^2=f(2)^3=f(2^3) I've messed around with knowing $f(3)=6$ and $f(2)=3$ but I am stuck.