  Let $(M,g)$ be a complete non-compact Riemannian manifold with the
$m$-dimensional Bakry-\'{E}mery Ricci curvature bounded below by a non-positive
constant. In this paper, we give a localized Hamilton-type gradient estimate
for the positive smooth bounded solutions to the following nonlinear diffusion
equation \[ u_t=\Delta u-\nabla\phi\cdot\nabla u-au\log u-bu, \] where $\phi$
is a $C^2$ function, and $a\neq0$ and $b$ are two real constants. This work
generalizes the results of Souplet and Zhang (Bull. London Math. Soc., 38
(2006), pp. 1045-1053) and Wu (Preprint, 2008).
