I was thinking about the fact that we can write certain irrational numbers as the result of an exponentiation of rational numbers, for example $\sqrt2=2^{0.5}$. My question is, can a irrational logarithm with rational operands be written as a power with a rational base and a rational exponent?
For example, let's take $y=\log_23$, is the following statement true?
$$\exists p, q\in\mathbb Q:p^q=y$$
If so, could you give a pair of values that makes this statement true?