Let's look at $p,q \in \mathbb P,\ p\ne q$. $\log_p(q)=a$ means $p^a=q$.
$a$ is irrational, because $a=\frac{r}{s}, \ r,s\in \mathbb N \Rightarrow p^{\frac{r}{s}}=q \Rightarrow p^r=q^s$ which violates the fundamental theorem of arithmetic.
Recently, in another context, the Gelfond Schneider Theorem was brought to my attention, and at first it seemed to clash with the notion that $a$ in the above example is irrational. I think I have found the resolution to my confusion.
$p$ being an integer is plainly an algebraic number. The Gelfond Schneider Theorem proves that for algebraic numbers $m,n; \ m\ne 0,1$; $m^n$ is transcendental.
Plainly $q$ is not transcendental. Thus, although $a$ is irrational, it cannot be an irrational algebraic number. The forced conclusion is that $a$ is transcendental, and so must be a great majority of logarithms. Moreover, the implication is that algebraic numbers raised to a transcendental power need not be transcendental, but can (in at least some cases) be integers or rational numbers. Gelfond Schneider does not necessarily hold when an exponent $n$ is transcendental.
My simple question is: Have I understood matters correctly?