Question

Prove that $\log {}_{2}3$ is an irrational number.

Answer 1

Suppose, if possible, that $lo{g}_{2}3$ is rational.
So we assume, $lo{g}_{2}3=\frac{p}{q}$
where $p$ and $q$ are positive integers having no factor in common. Then
$3=2^{p/q}or{3}^q=2^p$
which is impossible since $3^q$ is odd and $2^p$ is even. Hence $lo{g}_{2}3$ cannot be rational, that is it is irrational.