Rational Or Irrational?

Suppose $\log_{2}(7) \gt 0$ is rational, i.e., $\log_{2}(7) = \dfrac{p}{q}$ for some positive integers $p,q$ .

Then $\large 2^{\frac{p}{q}} = 7 \Longrightarrow (2^{\frac{p}{q}})^{q} = 7^{q} \Longrightarrow 2^{p} = 7^{q}$ .

But $2^{p}$ is even for any positive integer $p$ and $7^{q}$ is odd for any positive integer $q$ . Thus this last equality can never hold, and so by contradiction $\log_{2}(7)$ is $\boxed{\text{irrational}}$ .

$\log_{2} 7$ must be a irrational number because $\log_{2} 2^{n} = n , n > 0, n \neq 1$ , but $7$ is not a power of 2. So it is a irrational number.