Let a-f be integers g.t. 2 with $a < b < c < d < e < f$. Let
$$\ln def - \ln a b c = \alpha.$$
Let $\{p_i\}$ be the set of prime factors (with repetitions) in a,b,c. Let $\{q_i\}$ be the set of prime factors in d,e,f. Then we know that
$$ \ln def = \ln \prod q_i = \sum \ln q_i$$ and so for $p_i$ and a,b,c.
Then $$\sum \ln q_i - \sum \ln p_i = \alpha .$$
It is true I think that $def - abc \gg \ln def - \ln abc.$ My question is, can we make any quantitative statements $ \alpha = f (\beta)$ about
$$\sum q_i - \sum p_i = \beta$$ based on our knowledge of $\alpha$? It's tempting to say that $\alpha < \beta$, for example, but I don't see how to prove it.
Thanks for any suggestions/answers.