Let $p, \ell_1, \ell_2$ be distinct prime numbers, and $x_1, x_2 \in \overline{\mathbf{Q}}^\times$.
If $$ \frac{\log_p x_1}{\log_p \ell_1} = \frac{\log_p x_2}{\log_p \ell_2}, $$
does it follow that both ratios $\log_p x_i / \log_p \ell_i$ must be in $\mathbf{Q}$? (I know that both ratios must be either rational or transcendental, by Mahler's p-adic Gelfond--Schneider theorem.)
This is a typical case of the $p$-adic Four Exponentials conjecture. It is surely true, but the proof is beyond reach. If you add in a third prime (equality of $\log_p{x_i} / \log_p{\ell_i}$ for $i = 1,2,3$), then this becomes a case of the $p$-adic Six Exponentials theorem, a proof of which is available: [Serre J.-P.: "Dependance d'exponentielles $p$-adiques," Sem. Delange-Pisot-Poitou, vol. 7, no. 2, 1965/66, exp. 15].
The $p$-adic Four Exponentials conjecture is considered (and explicitly stated as Conjecture 6.0.3) in Chapter 6 of Tsanko Matev's thesis (https://epub.uni-bayreuth.de/1721/1/thesis.pdf).
