  We obtain a quantitative estimate of Bergman distance when $\Omega \subset
\mathbb{C}^n$ is a bounded domain with log-hyperconvexity index
$\alpha_l(\Omega)>\frac{n-1+\sqrt{(n-1)(n+3)}}{2}$, as well as the $A^2(\log
A)^q$-integrability of the Bergman kernel $K_{\Omega}(\cdot, w)$ when
$\alpha_l(\Omega)>0$.
