Admin4 writes anything it has to complain about to the logfile as configured under help/logging. The filename can be modified here. Usually, the file is named admin4.log and located in the user's home directory. The logging verbosity can be reduced/increased using the slider. If a python exception happens, it will be logged here as well.
A separate logging is performed for queries generated by Admin4. You can 'spy' on Admin4, observing which commands it sends to the server. If the querylog logfile is empty, no file is written.
Loglines may be marked and copied to the clipboard using Ctrl-C or the context menu.