We consider a damped plate equation on a smooth open bounded subset of R^d , or a smooth compact manifold with boundary, along with general boundary operators fulfilling the Lopatinskiȋ-Šapiro condition. The damping term acts on a internal region without imposing a geometrical condition. We derive a resolvent estimate for the generator of associated semigroup that yields a logarithmic decay of the energy of the solution to the plate equation. The resolvent estimate is a consequence of a Carleman inequality obtained for the bi-Laplace operator involving a spectral parameter under the considered boundary conditions. The derivation goes first through microlocal estimates, then local estimates, and finally a global estimate.