How can we proof that $2^{\sqrt 2}$ is irrational? How can we proof that $2^{\sqrt 2}$ is irrational?
I was trying with contradiction by taking $\frac xy=2^{\sqrt 2}$ but can't solve it by taking $log$ also. 

I think in this particular case if you take the log by base sqrt(2) and then change the base of logarithm to 2 you get a sort of proof of irrationality of sqrt(2) byunique factorisation problem. Just brainstorming though...
