Homepage › Solution manuals › Ivan Niven › An Introduction to the Theory of Numbers › Exercise 6.3.10 ($(\log 3)/(\log 2)$ is irrational)

An Introduction to the Theory of Numbers

Exercise 6.3.10 ($(\log 3)/(\log 2)$ is irrational)

Prove that $(\log 3) ∕ \log 2$ is irrational.

Answers

Proof. If $(\log 3) ∕ (\log 2)$ was rational, then

\frac{\log 3}{\log 2} = \frac{p}{q},

where $p, q$ are positive integers.

Then

3^{q} = 2^{p}, (p > 0, q > 0)

in contradiction with the unique factorization theorem. Therefore $\log (3) ∕ \log (2)$ is irrational. □

2025-07-21 09:38

Comments