TITLE: Irrationality of sum of two logarithms: $\log_2 5 +\log_3 5$
QUESTION [22 upvotes]: I try to prove that the number $$\log_2 5 +\log_3 5$$ is irrational. But I have no idea how to do it. Any hints are welcome.

REPLY [6 votes]: Questions of this form fall under the general heading of transcendental number theory. Conditional on a major conjecture in this field called Schanuel's conjecture, this number is not only irrational but transcendental. In fact, conditional on Schanuel's conjecture much more is true:

(Conditional) Theorem: The logarithms $\log 2, \log 3, \log 5, \dots$ of the primes are algebraically independent.

Essentially this means that any rational function of the logarithms of the primes is transcendental unless you can simplify it, as a rational function, to a rational number. The conclusion follows because your expression can be written $\frac{\log 5}{\log 2} + \frac{\log 5}{\log 3}$. 
Proof.  By unique prime factorization, the logarithms of the primes are linearly independent over $\mathbb{Q}$. If $p_1, p_2, \dots$ is an enumeration of the primes, then by Schanuel's conjecture it follows that $\mathbb{Q}(\log p_1, \log p_2, \dots \log p_k)$ has transcendence degree at least $k$, hence exactly $k$, for all $k$. $\Box$