OK, so I cheated and looked over at DBA.... :)
(The top two answers from 'RolandoMySQLDBA' and 'Derek Downey' are relevant and insightful.)
There are two types of 'log files' here, sadly. One is the logs of transactions applied to your MySQL databases. These files can be used to synchronize a MySQL slave to a MySQL master, or restore a database. As the top answer linked to in the page above shows, you should use only the MySQL commands (or meta-tools, such as Percona Toolkit) to "purge" these logs.
The other file, 'jaioyou-slow.log', is likely just a Slow Query Log, logging queries that take longer than a defined time to complete. This is a log file most similar to a typical *NIX log file, and therefore, can be truncated if needed (though you'll lose the performance statistics available in that file, obviously.)
You could use a useful tool like 'logrotate' to handle this (and only this) file, or, in an emergency, just do:
$ echo "" > /var/log/jaioyou-slow.log
- Your mysql configuration should put the MySQL Slow Query Log into a more appropriate *NIX log location, like /var/log/mysql/, rather than the /var/lib tree, to avoid confusion, and, perhaps, increase performance, if those directories are on separate disks.
- You'll need to have some kind of automation to purge thebinary logs from within MySQL (or third-party tools). I do not reccomend just deleting them with the operating system commands.