Hi,
Check the .ini file, it defines a logfile and a trace file:
[GLOBAL]
logging=some_file
[TRACE]
file=some_other_file
Make sure the given directories in front of the filename exists and are writable, and the thing should write logs (and traces, in case of serious problems). Most of the time.
And thus, the bad news: The agent does not always write logs when it has a problem. I found that in some cases, it simply crashes. In those cases there's two things you can do: debug it with strace or somesuch (advanced topic, complicated massively by the fact that it's Java), or check your installation trial-and-error style. E.g. check that you have the SAP libraries that are needed in the bin directory as per the documentation, and if you have set things like UC_EX_PATH_JOBREPORT set in the ini file, that the respective paths exist and are writable.
Hope this helps,
p.s. oh, and this: if you're starting the SAP agent from the service manager, try starting it from the shell directly, for testing. Sometimes it talks about it's existential issues on the console :)