Given $\{\log_ab \mid a,b\in \mathbb N, \mathrm{gcd}(a,b)=1,a,b≥3\}$ does the sum of any two $\log_ab$ form an irrational or rational number? Given $\{\log_ab \mid a,b\in \mathbb N, \mathrm{gcd}(a,b)=1,a,b≥3\}$ does the sum of any two $\log_ab$ form an irrational or rational number?

I know that $\log_ab$ is irrational, but does the sum of any two $\log_ab$ form an irrational or rational number?

e.g. Is $\{\log_3(5)+log_7(11) \in \mathbb Q\}$ or is $\{\log_3(5)+\log_7(11) \in \mathbb R \setminus \mathbb Q\}$.
Hint What is the identity of this operation? Is it an element in your set?
No, for instance, 

$log_5(7) - 2log_{25}(7) = \frac{log(7)}{log(5)} - 2\frac{log(7)}{log(25)} = \frac{log(7)}{log(5)} - \frac{2log(7)}{2log(5)} = \frac{log(7)}{log(5)}-\frac{log(7)}{log(5)} = 0$