How to compute an “effectively computable constant” in a formula of approximation of powers of $2$ and $3$

In his blog Terence Tao discusses the distance between powers of 2 and 3 and presents the following corollary:

Corollary 4 (Separation between powers of {2} and powers of {3})

• For any positive integers {p, q} one has

$$\displaystyle |3^p - 2^q| \geq \frac{c}{q^C} 3^p$$

for some effectively computable constants {c, C > 0} (which may be slightly different from those in Proposition 3).

What does he mean with "... effectively computable constants ... " ?

I've only a guess so far based on the inspection of the curve for $p$ and $q$ (=$N$ and $S$ in my usual notational style) using $p$ from the first hundred or so of the convergents of the continued fraction of $\log_2 (3)$ giving data for $p$ up to $1e175$ (only convergents where $2^q > 3^p$ are used).

From this I guess for instance $c=0.005$ and $C=1.01$. But those guesses might be much too crude.
I already presented an older guess in a MO-answer of mine but which seems even cruder.

So my question:

Q: How can one compute that constants?

pictures making my guess. Used only that cases where $2^S > 3^N > 2^{S-1}$ that means also from the original convergents of the continued fractions only each second one.

Image for the whole tested interval: Detail for the smaller leading interval: Detail for the smaller critical interval at $N \approx 1e166$: Picture rotated to make comparision better visible. Note that the labeling of the axes are now no more correct, and the apparent numbers $N$ are scaled due to rotation (note: the logs of all values were rotated using $\cos(),\sin()$ by $45$ deg). • I'm assuming... by reading the proof of this Proposition 3 (or, rather, of Baker’s theorem of which it is a corollary), that presumably is proven in not only a constructive way, but by describing algorithmically how to obtain these constants. – Clement C. Sep 6 '17 at 12:09
• @ClementC.- yes, this might be. But in any case: I'd like to see such a constructive computation or algorithm. I've come across such phrase "is effectively computable" frequently in the last weeks, but never found such computation been described - perhaps I'm missing some basic understanding here. – Gottfried Helms Sep 6 '17 at 12:58
• Well, have you had a look at the proof in question? – Clement C. Sep 6 '17 at 13:00
• @ClementC.:Hmm, I've read that blog-entry many times (not only this week) and never found any thing what would show me how I could actually do that "effective computation". Perhaps there's some blind spot on my side... – Gottfried Helms Sep 6 '17 at 13:07
• Oh, I am talking about the proof of Baker’s theorem... I don't know it, but as far as I can see it's not in that blog post. – Clement C. Sep 6 '17 at 13:08