If anyone could me any advice/direction of how to approach this problem, I'd greatly appreciate it. I couldn't find anything on the internet that looked or was similar to proving this type of proof.
The fact that that there are irrational numbers a,b such that ab is rational was proved in Problem 1.5. Unfortunately, that proof was nonconstructive: it didn’t reveal a specific pair, a,b, with this property. But in fact, it’s easy to do this:
We know √2 is irrational, and obviously a=3. Finish the proof that this a,b pair works, by showing that 2 log base 2 of 3 is irrational.