Let Pt be the diffusion semigroup generated by L:=Δ+∇V on a complete connected Riemannian manifold with Ric≥−(σ2ρo2+c) for some constants σ, c>0 and ρo the Riemannian distance to a fixed point. It is shown that Pt is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided −HessV≥δ holds outside of a compact set for some constant $\delta >(1+\sqrt{2})\sigma \sqrt{d-1}$ . This indicates, at least in finite dimensions, that Ric and −HessV play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.