Question

Show that ${\log }_{2}7$ is an irrational number.

Answer 1

Let ${\log }_{2}7$ is a rational number.

Then, ${\log }_{2}7=\frac{p}{q}$

$\Rightarrow 7=2^{p/q}$

$\Rightarrow 7^q=2^p$
which is not possible for any integral values of $p$ and $q$.
Hence, our assumption is wrong.

Hence, ${\log }_{2}7$ is not rational.