A Voevodsky motive associated to a log scheme

For each fs log scheme $(X,{\mathcal{M}}_X)$ over a field $k$ we construct a geometrical Voevodsky motive $[X{]}^{log}\in D{M}_{gm}(k,\mathbb{Q})$. We prove that, for $k=\mathbb{C}$, the Betti realization of $[X{]}^{log}$ is the log Betti cohomology of $(X,{\mathcal{M}}_X)$. We give applications to motivic tubular neighborhoods, limit motives and the monodromy filtrations.