  In this paper, we verified the critical conjecture in our previous work
\cite{wang2023wave} on two-dimensional hyperbolic space, that is, concerning
nonlinear wave equations with logarithmic nonlinearity, which behaves like
$\left(\ln {1}/{|u|}\right)^{1-p}|u|$ near $u=0$, on hyperbolic spaces, we
demonstrate that the critical power is $p_c(2)=3$, by proving global existence
for $p>3$, as well as blow up for $p\in (1,3)$.
