$|2^x-3^y|=1$ has only three natural pairs as solutions
As the title says:
Find all positive integers k and l such that $3^k-2^l=1$
In previous questions it is proved that $2^l+1$ is divisible by $27$ iff it is divisible by $19$, though I'm not sure how that is meant to help.
I've brute forced it, and got that $k = l = 1$ works, as does $k = 2$ and $l = 3$, but I can't figure out how to prove that these are the only solutions.
Any help would be much appreciated