This paper is mainly devoted to estimate the logarithmic Sobolev (abbrev. L.S.) constant for diffusion operators on manifold or in Rd. In most cases, we study the lower bounds but a gener- alization to [9; Theorem 1] for the upper bound is also presented (Theorem 1.5). Based on a simple observation (due to [5]) of the comparison between the L.S. constants for different potentials, the pow-erful Bakry-Emery criterion for the L. S. inequality is improved considerably in the paper, especially for the manifolds with non-positive sectional curvatures (Theorem 1.3 (1)). In terms of our notation: