${\log }_{2}7$ is

Let ${\log }_{2}7$ be a rational number equal to $\frac{m}{n}$, where

$m,n\notin N$ and they do not have a common factor. Then

${\log }_{2}7=\frac{m}{n}\Rightarrow 7=2^{m/n}\Rightarrow 7^n=2^m$

Clearly, this is impossible, because $7^n$is an odd natural number and $2^m$ is an even natural number. Hence,$Iog{2}^7$ is an irrational number