So, I've been running in both stacky circles and logarithmic circles and I've been wondering: is there a definition of log stack that is "useful"? I can imagine two such definitions:
1) A log stack is a stack along with an effective divisor (could be useful for studying moduli of smooth curves, then)
2) A log stack is a stack over the category of log schemes
Is either one in standard use? Are they equivalent?
This question was answered in Martin Olsson's thesis ( http://math.berkeley.edu/~molsson/thesis.ps ). He gives sufficient conditions for a fibered category on the category of log schemes to arise from "a stack with a log structure", which I think is more or less what Charles intended by Option 1.
Option 1 is definitely not the definition - in the scheme setting, it doesn't yield log structures on the divisors in question, and you don't get log points. A.J. mentioned that you probably meant to say, "stack equipped with a log structure". You can find a definition in section 5 of Martin Olsson's paper Logarithmic geometry and algebraic stacks. His web page doesn't have it, but you can find it on Google Scholar.
Option 2 works also. Fine log schemes over a base fine log scheme form an algebraic stack over the underlying base scheme. You can do your log stack theory over this. See Corollary 5.8 in Olsson's paper comparing the two options.
$\DeclareMathOperator\Log{Log}$From Gilliam - Logarithmic stacks and minimality:
A category fibered in groupoids (CFG) over schemes with a map to Olsson's stack $\Log$ induces a CFG over log schemes. This is fully faithful, but there are loads of other CFG's over log schemes that do not come from this process.
To determine which ones do, note that if a CFG has a map $X \to \Log$, any scheme mapping to $X$ may be given the pulled-back "minimal" log structure. Among the log schemes mapping to $X$ with the same underlying scheme, there's a final one whose map to $X$ is strict. There's a categorical way to describe and detect minimality.
The map $X \to \Log$ could parametrize a Cartier divisor as you suggest, or more generally any log structure. Log structures are essentially given by systems of line bundles with sections with some relations among their tensor powers (you need "quasi-integrality" here).
