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



Equation in integers of irrational degree
<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.


