To resonate with Henry Cohn's comment, Schanuel's conjecture implies that the natural logarithms of the primes are algebraically independent over $\mathbb{Q}$. The problem may well be very hard (i.e. open).

To resonate with Henry Cohn's comment, Schanuel's conjecture implies that the natural logarithms of the primes are algebraically independent over $\mathbb{Q}$. In particular, the statement in the original post is probably true, but proving it might be out of reach at the moment.

To resonate with Henry Cohn's comment, Schanauel's conjecture implies that the natural logarithms of the primes are algebraically independent over $\mathbb{Q}$. The problem may well be very hard (i.e. open).

To resonate with Henry Cohn's comment, Schanuel's conjecture implies that the natural logarithms of the primes are algebraically independent over $\mathbb{Q}$. The problem may well be very hard (i.e. open).