# More questions about log structures

I now have some more questions regarding the role of log structures in moduli problems (you can assume that the moduli problem is the compactification of $n$-marked genus $g$ smooth projective curves for simplicity):
1. It seems that one of the mantras of the subject is that outside of the boundary, the objects have a unique log-structure. In terms of the example moduli problem I gave, that $n$-marked smooth projective curves of genus $g$ have unique log-structures. In what sense is this true? It doesn't seem literally true to me. Surely they must mean that they have unique log-structures such that they satisfy some property, right? If you can enlighten me about the essence of this mantra, please do!
2. One of the strengths of log-structures, evidently, is that in the degenerations, they give a unique deformation. So in the example, if we had a stable $n$-marked curve of genus $g$ with a log-structure, there there would be a unique way to extend it to a complete DVR. My question is: what is the virtue of log-structures as opposed to deformation data? Why not instead of a log-structure attached to each (possibly semi-stable) curve, just add some data that will say how it deforms over a complete DVR? Would it be fair to say that log-structures is the natural way to encode this deformation data? Or perhaps there is an extra virtue? I'm confused about this.