How active should I expect my Jelly Bean system log file to be?
Not very active if you use apps like aLogcat. The permissions system regarding to logs was changed with Jelly Bean: The permission to read logs "READ_LOGS" is now systemOrSignature, which means that an app that is not signed with the ROM/system key can only view its own log. You can still view the full android log with adb logcat from the SDK.

I think that this was a long overdue change in the log system.