How to prove $1/ \log a + 1/ \log b$ for rational $a$ and $b$ is a transcendental number?

Question

I know how to prove $\log a$ for rational $a$ is transcendental, because if it were algebraic it would imply $e$ is algebraic as well (namely if $\log a = b, e = a^{1/b}$), and I can prove $\log a + \log b$ for rational $a$ and $b$ is transcendental because that's just $\log ab$, and the same proof follows. But I don't know how to prove $\frac{1}{\log a} + \frac{1}{\log b}$ is transcendental, because it boils down to if $(\log a)(\log b)$ is transcendental for rational $a$ and $b$ and I don't know how to prove that. I can prove the simplest case, for when $a = b$, but I don't know how to expand that proof.

Answer 1

doesn't work for $b=1/a$

.................................