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

Abbie Mcgrath

Abbie Mcgrath

Answered question


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

Answer & Explanation

Amina Hall

Amina Hall

Beginner2022-01-23Added 11 answers

Your identity gives:
where xQ 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 1x and xg are rational numbers with the same denominator, g has to be one. So x=12 and g=1 is the only solution.

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