Show the "trivial bound" $\left|\frac{\log{2}}{\log{3}}-\frac{p}{q}\right|\ge c \frac{1}{2^q}$ holds for $c > 0$ and integers $p, q$ with $q$ positive

Question

Given the identities:

1. $3^p - 2^q = 3^p (1 - 3^{q\left(\frac{\log{2}}{\log{3}} - \frac{p}{q}\right)})$ for all integers $p, q$ with $q$ positive.

2. $\left|3^p - 2^q\right| \ge 1$ for all positive integers $p, q$.

Show there exists a constant $c > 0$, such that

$\left|\frac{\log{2}}{\log{3}} - \frac{p}{q}\right| \ge c \frac{1}{2^q}$

for all integers $p, q$ where $q > 0$.

(Paraphrased from Proposition 5 (Trivial bound) in Terence Tao's blog article.)

I assumed it would just involve some algebra, but I couldn't solve it after scribbling a page full of notes.