Are there any algebraic irrational numbers in $\{log_xyx,y\in\mathbb{N},x,y\geq2\}$?
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http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem, 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. 

