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In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds $(M,g)$ for the following general heat equation $$u_t={\Delta }_{V}u+aulogu+bu$$ where $a$ is a constant and $b$ is a differentiable function defined on $M\times [0,\infty )$. We suppose that the Bakry-Émery curvature and the $N$-dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently.