In this paper, the exponential ergodicity in relative entropy is derived for non-degenerate SDEs with multiplicative noise under partially dissipative condition. Instead of the important tool of the log-Sobolev inequality, we adopt the hypercontractivity of the associated semigroup so that we do not require the famous Bakry-Emery curvature lower bound condition. The result greatly improves the ones in the additive noise case.
