This paper studies the distributional asymptotics of the slowly changing sequence of logarithms (logb n) with b ∈ N \ {1}. It is known that (logb n) is not uniformly distributed modulo one, and its omega limit set is composed of a family of transl√ated exponential distributions with constant log b. An improved upper estimate log N/N 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 conver- gence (log N/N) 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 (log N/N) as well, which verifies that an up- per bound for this rate derived in [Ohkubo, Y.—Strauch, O.: Distribution of leading digits of numbers, Unif. Distrib. Theory, 11 (2016), no.1, 23–45.] is sharp.