Sometimes, it is not easy to choose a compactification of a moduli space, especially if the objects being parametrized are complicated - one may find that a choice of degenerate structure is too permissive, and is parametrized by superfluous components. One reason why log structures are useful is that they often yield parsimonious degenerations of structures (and thereby, natural compactifications of moduli spaces).

A relatively simple a posteriori example comes from the moduli of smooth pointed curves. To compactify, you can generalize smoothness to allow at most nodal singularities (Deligne-Knudsen-Mumford), or you can add a (fs) log structure, and consider the moduli problem of log-smooth integral pointed curves (F. Kato). More generally, you can compactify the moduli of smooth twisted curves using twisted stable curves (Abramovich-Vistoli), or twisted log curves (Olsson). In the case of log curves, one finds that a certain "balanced" condition appears automatically, and excludes curves whose nodes don't have matching orbifold structure.

I've never seen the map $\alpha$ referred to by names like logarithm and exponential, but the notion of logarithm is appropriate when considering differentials. If you take the affine line with the natural chart $\mathbb{N} \mapsto \mathbb{C}[\mathbb{N}] = \mathbb{C}[t]$, the sheaf of log differentials contains a section of the form $\frac{dt}{t} = d\log t$. Similarly, if you have a log-smooth curve, its sheaf of relative differentials is the dualizing sheaf, which is made of ordinary differentials on the (schematically) smooth locus, and has at most log poles at nodes, where the residues on the two pieces sum to zero.

I found the introduction of Kato-Usui (Classifying spaces of degenerating mixed Hodge structures) rather inspirational.