  In this paper, we prove Hamilton's Harnack inequality and the gradient
estimates of the logarithmic heat kernel for the Witten Laplacian on complete
Riemainnian manifolds. As applications, we prove the $W$-entropy formula for
the Witten Laplacian on complete Riemannian manifolds, and prove a family of
logarithmic Sobolev inequalities on complete Riemannian manifolds with natural
geometric condition.
