Gaia Sky provides a couple of ways of accessing system logs.
Gaia Sky always saves the log of the last session to
$GS_DATA/log/gaiasky_log_lastsession.log (check where
$GS_DATA is here). If you need to check the full log of your last session, you can always find it there.
If Gaia Sky crashes, a crash report, together with a full session log to
$GS_DATA/crashreports (check where
$GS_DATA is here). Files with the form
gaiasky_crash_[date].txt are crash reports, while files with the form
gaiasky_log_[date].txt are full session logs. You can attach these whenever a crash happens and you want to submit a bug report to our buck tracker.