I'm not aware of a technique, to write all java exceptions to the syslog.
If you start from commandline, exceptions will be written to stderr, aka the console, as long as you don't redirect them.
If a script starts the program, it may redirect the errormessages to the syslog.
If the exception isn't caught, you would find a stacktrace in the syslog - if you only find single messages, they were caught, I guess.
Of course the softwaredeveloper might have used a logging-mechanism to log errors, and might have specified syslog.log in the code. But again, I would expect a StackTrace for uncaught exceptions.
You could try to find out which program threw the exception to find out more - I would expect it's name in the logfile or search for date/time ...
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