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.