Let $u:M \times I \to (0,\infty)$ be a positive heat solution on a fixed Riemannian manifold $(M,g)$, and define $f:M\times I\to\mathbb{R}$ by $f(p,t)=\log(u(p,t))$. Then
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\begin{align*}
\partial_t f = \Delta_g f + |\nabla f|^2.
\end{align*}