  We consider the stochastic differential equation $$ \left\{ \begin{array}{lc}
dX(t)=[AX(t)+F(X(t))]dt+C^{1/2}dW(t), & t>0;\\ X(0)=x \in \mathcal{X};
\end{array}\right. $$ where $\mathcal{X}$ is a Hilbert space, $\{W(t)\}_{t\geq
0}$ is a $\mathcal{X}$-valued cylindrical Wiener process, $A, C$ are suitable
operators on $\mathcal{X}$ and $F:{\rm Dom}\,(F)\subseteq \mathcal{X}\to
\mathcal{X}$ is a smooth enough function. We establish a logarithmic Harnack
inequality for the transition semigroup $\{P(t)\}_{t\geq 0}$ associated with
the stochastic problem above, under less restrictive conditions than those
considered in the literature. Some applications to these inequalities are also
shown.
