This question arose as an attempt to answer the following question Relaxed Collatz 3x+1 conjecture. I wanted to show that there is a solution of equation $2^{k}=3^{z}(2n+1)-1$ for each $n\geq 2$, where $k,z,n\in\mathbb{N}$. But even a special case has put me in a dead end.

Is there infinitely many solutions of equation $2^k=3^z-1$, when $z\rightarrow \infty$? First solution: $2^3 =3^2 -1$.