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Are there any algebraic irrational numbers in $\{log_xy|x,y\in\mathbb{N},x,y\geq2\}$?

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up vote 3 down vote accepted,

Suppose your set produces an algebraic irrational number $ \log_xy = z,$ then $ x^z = y $ is transcendental by the theorem . But $y$ is a natural number, which are algebraic by nature. Thus we have arrived at a contradiction.

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