  In this paper, we consider the gradient estimates of the positive solutions
to the following equation defined on a complete Riemannian manifold $(M, g)$
$$\Delta u + au(\log u)^{p}+bu=0,$$ where $a, b\in \mathbb{R}$ and $p$ is a
rational number with $p=\frac{k_1}{2k_2+1}\geq2$ where $k_1$ and $k_2$ are
positive integer numbers. we obtain the gradient bound of a positive solution
to the equation which does not depend on the bounds of the solution and the
Laplacian of the distance function on $(M, g)$. Our results can be viewed as a
natural extension of Yau's estimates on positive harmonic function.
