Elementary Number Theory: Proving log$_2$($3$) is irrational using the Fundamental Theorem of Arithmetic.

Problem: Prove $log$$_2$($3$) is irrational using the Fundamental Theorem of Arithmetic.

What I have so far:

Proof: Suppose $log$$_2$($3$) $\in$ $\mathbb Q$

Then there are $p,q$ $\in$ $\mathbb Z$ , $q$$\neq$$0$ s.t. $log$$_2$($3$)=$\frac pq$

Then $2$$^p$ $=$ $3$$^q$

This is where I get stuck because I'm not sure how to incorporate the Fundamental Theorem of Arithmetic.