Hi,
The log (don't confuse that with reports, I'll get back to those in a moment) primarily keeps track of problems avast ecounters with itself or with the system (problems installing, for example). If it's running smoothly then it's normal for the logs to be empty, or nearly empty.
Virus-detection activity (whether via the resident protection providers or an on-demand scan) is recorded in the reports, if you have reports enabled. You didn't mention whether you're using the Home or Pro version -- with Home (which I use), you can access those manually, they're ordinary txt files in the Data/Report folder under the Avast folder.
I believe in the Pro version (and of course I'm open to correction on this) the reports can be accessed more easily right from avast's own menu in its interface.
Best,
Mike