To reduce the size of big log files use truncate - e.g.
sudo truncate -s 0 /var/log/daemon.log
just in case the service does not recreate the file automatically. If you just delete the file it can make things worse.
What does df -h tell you?
You can use journalctl commands to view the entries being passed to rsyslog by systemd.
Are you using /var/log on tmpfs?