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.)



Transcendence of a ratio of p-adic logarithms
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>).


