# Prove that a logarithm is irrational I’m stuck with the following problem: Prove that log_(2) 3 in bbb{R} - bbb(Q) Thanks in advance!

Cale Terrell 2022-10-21 Answered
Prove that a logarithm is irrational
I’m stuck with the following problem:
Prove that ${\mathrm{log}}_{2}3\in \mathbb{R}-\mathbb{Q}$
Thanks in advance!
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## Answers (1)

Liam Everett
Answered 2022-10-22 Author has 16 answers
Suppose that log_2 3 is rational, i.e.
${\mathrm{log}}_{2}3=\frac{n}{m}$
where m,n are positive integers.
Then, we have
${\mathrm{log}}_{2}3={\mathrm{log}}_{2}{2}^{\frac{n}{m}}⇒3={2}^{\frac{n}{m}}⇒{3}^{m}={2}^{n}.$
Here, the LHS is odd and the RHS is even, which is a contradiction.
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