TITLE: Proving the Irrationality of this Number
QUESTION [19 upvotes]: I found this problem on Math.SE: 

Prove that $\log_35+\log_25$ is irrational.

https://math.stackexchange.com/q/986227/173397. 
I labored on it for a few days, and couldn't find an algebraic solution- I'm not even sure if such a solution exists. All I was able to do was prove that both components were irrational by themselves (as opposed to their sum). I am wondering if anyone has seen this problem before, and/or if anyone knows a solution. If so, I could really use a hint. 
So far, using the Fundamental Theorem of Arithmetic (i.e., all integers have a unique prime factorization) hasn't helped me the way one would use it to show that the individual components are irrational. 
Thank you in advance.

REPLY [8 votes]: 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.