This paper studies the distributional asymptotics of the slowly changing sequence of logarithms $(\log_bn)$ with $b\in\mathbb{N}\setminus\{1\}.$ It is known that $(\log_bn)$ is not uniformly distributed modulo one, and its omega limit set is composed of a family of translated exponential distributions with constant $\log b.$ An improved upper estimate $\left(\sqrt{\log N}/N\right)$ is obtained for the rate of convergence with respect to (w.r.t.) the Kantorovich metric on the circle, compared to the general results on rates of convergence for a class of slowly changing sequences in the author's companion in-progress work. Moreover, a sharp rate of convergence $\left(\log N/N\right)$ w.r.t. the Kantorovich metric on the interval $[0,1]$, is derived. As a byproduct, the rate of convergence w.r.t. the discrepancy metric (or the Kolmogorov metric) turns out to be $\left(\log N/N\right)$ as well, which verifies that an upper bound for this rate derived in [Y. Ohkubo and O. Strauch, Distribution of leading digits of numbers, Unif. Distrib. Theory, $\textbf{11}$ (2016), no.1, 23--45.] is sharp.