Question: > 
> Prove that $\log\_2 3$ is irrational
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Seemingly simple homework assignment. Was never the best with logarithms, how would I go about proving?

Answer: 
You can't because it isn't. Assume $\log\_2(3)=\frac pq$, then$ 3=2^{\frac pq}, 3^q=2^p$ so???
Assume, $x := log\_2(3) = \frac{a}{b}$ 

As x > 0 , we can assume that a and b are positive integers.

Now $$2^x=3$$

leads to $$2^\frac{a}{b}=3$$

Therefore $$2^a=3^b$$

If a and b are positive, the left side is even, the right side odd.
 A clear contradiction.