Refinements of the classical Hermite-Hadamard (or Jensen-Hadamard) inequality for log-convex functions have been obtained recently by S. S. Dragomir and B. Mond [Demonstr. Math. 31, No. 2, 355-364 (1998; Zbl 0912.26009)]. In this paper, another refinement is given in the case of differentiable log-convex functions. Some remarks on applications for special means of two arguments are also pointed out.