Abbie Mcgrath

2022-01-22

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

Amina Hall

$1-x={x}^{g}$
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.

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