In addition to log files that are created for specific checks/processes in the <GeoMonitor Install Location> \Scripts folder, the GeoSystems Monitor also records important application events information in the Windows Event Log (Computer Management dialog > Event Viewer > Windows Logs > Application).  GeoSystems Monitor application events in the Windows Application Event Logs can be filtered by the Event Source = 'GeoMonEngine'.  See snapshot below:

When debugging is turned on in the GeoMonitor dashboard, a more verbose set of logs are written to the Event Log. 

Clicking the LOG icon will open up a page with the different log files, then you can open up the log file needed with the associated check.