If x is rational, can log(1−x)logx be algebraic?
If x is rational, can be algebraic?
Answer & Explanation
Beginner2022-01-23Added 11 answers
Your identity gives:
where trivially gives that the LHS is a rational number. If is not a rational number, the Gelfond-Schneider theorem gives that the RHS is a trascendental number, contradiction.
So has to be a rational number. But in order that and are rational numbers with the same denominator, has to be one. So and is the only solution.
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