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 $[X]^{log}\in DM_{gm}(k,\mathbb Q)$. 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.