Four positive integers $a,b,c,d>1$ satisfy $\log a \log b = \log c \log d$. Is necessarily $\frac{\log a}{\log c} \in \mathbb{Q}$ or $\frac{\log a}{\log d} \in \mathbb{Q}$?
I tried to use Gelfond–Schneider theorem and Lindemann–Weierstrass theorem from transcendence theory but it didn't work.
Yes, but I can only do it with something heavier, namely Schanuel's conjecture, or more precisely the following corollary of it.
Theorem (conditional on Schanuel's conjecture): Let $p_1, p_2, \dots$ be the primes. Then $\log p_1, \log p_2, \dots$ are algebraically independent over $\mathbb{Q}$.
Proof. By unique prime factorization, $\log p_1, \log p_2, \dots$ are linearly independent over $\mathbb{Q}$. Now apply Schanuel's conjecture with $z_i = \log p_i$ for all $n$. $\Box$
If $a$ is any positive integer, then expanding out $\log a$ using the prime factorization of $a$ allows us to write it as a homogeneous linear polynomial in the variables $\log p_i$ (with rational coefficients). Hence the identity
$$\log a \log b = \log c \log d$$
asserts that some homogeneous quadratic polynomial in the variables $\log p_i$ (with rational coefficients) has two factorizations into linear factors, which must necessarily agree up to a permutation and scalar multiplication because the ring $\mathbb{Q}[\log p_i]$, being a polynomial ring, is a UFD.
