If x is rational, can log(1−x)logx be algebraic?

Abbie Mcgrath

Answered question

2022-01-22

If x is rational, can $\frac{\mathrm{log}(1-x)}{\mathrm{log}x}$ be algebraic?

Answer & Explanation

Amina Hall

Beginner2022-01-23Added 11 answers

Your identity gives:
$1-x={x}^{g}$
where $x\in \mathbb{Q}$ trivially gives that the LHS is a rational number. If $g$ is not a rational number, the Gelfond-Schneider theorem gives that the RHS is a trascendental number, contradiction.
So $g$ has to be a rational number. But in order that $1-x$ and $x}^{g$ are rational numbers with the same denominator, $g$ has to be one. So $x=\frac{1}{2}$ and $g=1$ is the only solution.