  We consider the log-perturbed Br\'ezis-Nirenberg problem on the hyperbolic
space
  \begin{align*}
  \Delta_{\mathbb{B}^N}u+\lambda u +|u|^{p-1}u+\theta u \ln u^2 =0, \ \ \ \ u
\in H^1(\mathbb{B}^N), \ u > 0 \ \text{in} \ \mathbb{B}^N.
  \end{align*}
  and study the existence vs non-existence results. We show that whenever
$\theta >0,$ there exists an $H^1$-solution, while for $\theta <0$, there does
not exist a positive solution in a reasonably general class. Since the
perturbation $ u \ln u^2$ changes sign, Pohozaev type identities do not yield
any non-existence results. The main contribution of this article is obtaining
an ``almost" precise lower asymptotic decay estimate on the positive solutions
for $\theta <0,$ culminating in proving their non-existence assertion.
