  In light of the exponential decay of solutions of linear wave equations on
hyperbolic spaces $\mathbb{H}^n$, to illustrate the critical nature, we
investigate nonlinear wave equations with logarithmic nonlinearity, which
behaves like $\left(\ln {1}/{|u|}\right)^{1-p}|u|$ near $u=0$, on hyperbolic
spaces. Concerning the global existence vs blow up with small data, we expect
that the problem admits a critical power $p_c(n)>1$. When $n=3$, we prove that
the critical power is $3$, by proving global existence for $p>3$, as well as
generically blow up for $p\in (1,3)$.
