Exercise 59Number theory › IrrationalityHigh schoolAdded 3 Nov 2026

Irrationality of log₂ 3

#logarithm#irrationality#base change

Statement

Show that $\log_2 3$ is irrational.

Hint

If $\log_2 3=p/q$ , then $2^p=3^q$ .

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Assume. $\log_2 3=p/q$ with $p,q\in\mathbb{N}^*$ .

Then. $2^{p/q}=3$ , so $2^p=3^q$ .

Contradiction. Unique factorization: a power of $2$ cannot equal a power of $3$ (unless both are $1$ , but $p,q>0$ ).

Conclusion. $\log_2 3$ is irrational.