  The logarithmic multiplicative group is a proper group object in logarithmic
schemes, which morally compactifies the usual multiplicative group. We study
the structure of the stacks of logarithmic maps from rational curves to this
logarithmic torus, and show that in most cases, it is a product of the
logarithmic torus with the space of rational curves. This gives a conceptual
explanation for earlier results on the moduli spaces of logarithmic stable maps
to toric varieties.
