I'm currently reading a thesis that uses log-structures. I should mention that this is my first encounter with them, and the thesis (as well as my expertise) is scheme-theoretic (in fact stack-theoretic) and so the original geometric motivations are lost on me.

Here is my meek understanding. For any scheme, we can give a log-structure. This is a sheaf , $M$, fibered in monoids, on the etale site over a scheme $S$; together with a morphism of sheaves fibered in moinoids $\alpha:M\rightarrow O_S$ such that when it is restricted to $\alpha^{-1}(O_S^{\times})$ it is an isomorphism.

This $\alpha$ is called the exponential map, and for any $t\in O_S(U)$ (for some $U$), a preimage of it via $\alpha$ is called $log(t)$($\in M(U)$).

I am curious about a few things, and puzzled about others. First, in terms of the notation, surely it's no coincidence that these are called exponential maps and log-structures. What is the geometric motivation for it?

Second, these come up in the thesis I'm reading in the context of tame covers. I am puzzled about what, precisely, log-structures contribute. It seems to me, in extremely vague terms (commensurate with my understanding), that the point of log-structures in this context is that if you add this *extra information* to tame covers it somehow helps you construct *proper* moduli spaces of covers.

On top of everything I'm also confused about the role of `minimal log-structures' in all of this.

In conclusion, if you can say anything at all about the motivations of log-structures in the geometric setting, or more importantly in the context of tame covers, I would extremely appreciate it. The plethora of notationally different texts on the subject is making it hard to understand the gist of what's going on.

Also, if you have examples that I should have in mind when thinking about it, that would be ideal.