You didn't say which Linux distro you are using, but regardless, you very likely have a package named logrotate installed. It takes care of compressing old logs and deleting older logs to take care of this very problem.
If your /var partition is too small, you may want to edit /etc/logrotate.conf and change the line that says
or similar. This line determines how many weeks of logs are kept. Also, be sure that the line that says
is uncommented so that old logs are compressed.
To learn more, see the command
Those logs can be crucially valuable to figuring out why something doesn't work, so it's unwise to delete them earlier than necessary. It's interesting to note how many people post questions on sites like this without first looking at their logs.