In his blog, Terence Tao discussed the lower bound of $\vert 3^p - 2^q \vert$ in the following corollary.
Corollary 4 (Separation between powers of $2$ and powers of $3$) For any positive integers $p$, $q$ one has
$$|3^p - 2^q| \geq \frac{c}{q^C} 3^p$$
for some effectively computable constants $c, C > 0$ (which may be slightly different from those in Proposition 3).
This appears to be a well-established result. How do I derive this result from Baker's theorem?
Write $$ \Lambda = q \log 2 - p \log 3. $$ Then $$ 2^q - 3^p = 3^p \cdot (e^\Lambda -1) $$ and the problem amounts to finding a lower bound upon $|e^\Lambda -1|$ that is $\gg q^{-C}$, for some positive $C$. To do this, just note that $|e^z-1|$ is close to $|z|$ for $|z|$ small (you can easily make this explicit). It follows that all you want is a suitably good lower bound upon $|\Lambda|$, which follows from Baker's theorem (though such a bound for $2$ logarithms goes back rather further, to Gelfond).