We mainly introduced an extended Ricci flow introduced by Bernhard List in 2005.In the first part, we give a log Sobolev inequality which is equivalent to a Sobolev inequality.In the second part we mainly establish some point-wise gradient estimates for positive solutions of the conjugate heat equation.First we give a gradient estimate for the fundamental solutions of the conjugate heat equation under an extended Ricci flow system.Second, we improve gradient estimates to all positive solutions with curvature assumption and without curvature assumption respectively.Finally, under the assumption of the lower bound on the Ricci curvature, we establish a local version of gradient estimate similar to the the Li-Yau estimate for the linear heat equation.And also, we show that every gradient estimate we derived implies a corresponding harnack inequality.