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. 

Yes, it is famous Gelfond-Schneider theorem.
If $a$ and $b$ are algebraic numbers with $a \neq 0$, $1$ and $b$ irrational, then any value of $a^b$ is a transcendental number.
