> My syslog works and become huge file. Can I delete it?
Sure. It is only a history of events that took place. As a sysadmin, we gzip compress the logs & save them, but if you have no reason to hold on to those events then there should be no problem deleting it.

I usually delete & recreate the file so that the syslog service will continue to function normal. I have seen it misbehave if the syslog is missing.
rm /var/log/syslog
cat /dev/null > /var/log/syslog