I am interested in compactification of the moduli space of elliptic curves, and I heard that Log geometry is very important for the problem. 
I am developping the same technique for quantum geometry. 


My question was that:


1-Why Log structure can give us a better way to understand degeneration of Elliptic curves? What's the motivation behind?


Is that right that log structure gives us a way to embed the scheme locally into affine space, and the degeneration happens in the space?
Like the log structure on $N^2\rightarrow k[x,y]/x.y=0.$, which is fibered over $N\rightarrow k$. In that case, the fourier transform maps $N^2$ into $A^2$, which is the embeding of the nodal curve.


2-Does it make sense to define log-group, i.e. a group with a log structure.


3-Does it make sense to define log structure on a stack which is not algebraic? 


Reference for Log geometry: <http://www-personal.umich.edu/~satriano/logcurves.pdf>



Log structure and degeneration
1. Read the introduction in the Kato-Usui book. It's got some nice motivating examples, including a degenerating family of elliptic curves with pictures.
2. There is no problem with defining a group object in the category of log schemes.
3. This depends on what you want to do with such a structure, i.e., what properties you need. Maybe you should familiarize yourself with the algebraic case first (e.g., in Olsson's paper *Logarithmic geometry and algebraic stacks*).


Finally, Matt's notes are a nice introduction, but I hesitate to call them a reference. You might want to look at Ogus's book or Kato's original paper.


