Question

Prove that ${\log }_{2}7$ is irrational.

Answer 1

Assume that ${\log }_{2}7$ is rational,

$\Rightarrow {\log }_{2}7=\frac{p}{q}$ where $p,q\in I$.

Changing to exponential form,

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

$\Rightarrow 7^q=2^p$.

It can be seen that the L.H.S is odd and R.H.S is even.

Hence there is a contradiction and therefore ${\log }_{2}7$ is irrational.

Therefore, hence proved.