[Math] Can transcendental to the power transcendental be rational

Question

Can a transcendental number to the power of a transcendental number be a rational number?

Answer 1

This is a bit easier to see the other way around. Let $q\in\Bbb Q$ be positive ($q\ne 1$), and consider the map $x\mapsto q^x$. This is a bijection from $\Bbb R$ to $(0,\infty)$, and if you restrict it to the transcendental numbers, you still have an uncountable image. That image must contain uncountably many transcendental numbers, because we only have countably many algebraic numbers.

Thus, we have $q^{t_1}=t_2$ for some transcendental numbers $t_1$ and $t_2$. The number $1/t_1$ is also transcendental, and $t_2^{1/t_1}=q$.

In fact, this shows that, if a positive rational number is not $1$, then it can be written as a transcendental power of a transcendental number in uncountably many different ways.