Rational curves in the logarithmic multiplicative group.

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.