It converges for $q<1$ and diverges for $q\ge 1$. You can separate the interval into the ranges $[e,1/\sqrt c]$ and $[1/\sqrt c,\infty)$. On the first range, the exponential term is essentially 1. On the second range, the logarithm is essentially constant.

It converges for all $q$. You can separate the interval into the ranges $[e,1/\sqrt c]$ and $[1/\sqrt c,\infty)$. On the first range, the exponential term is essentially 1. On the second range, the logarithm is essentially constant.