  We define higher pro-Albanese functors for every effective log motive over a
field $k$ of characteristic zero, and we compute them for every smooth log
smooth scheme $X=(\underline{X}, \partial X)$. The result involves an inverse
system of the coherent cohomology of the underlying scheme as well as a
pro-group scheme $\mathrm{Alb}^{\log}(X)$ that extends Serre's semi-abelian
Albanese variety of $\underline{X}-|\partial X|$. This generalizes the higher
Albanese sheaves of Ayoub, Barbieri-Viale and Kahn.
