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.