How do I output MySQL logs to syslog?
This is MUCH more simply done this way:

cd to mysql folders:

mkfifo mysql-general.log


in my.cnf tell it:

[mysqld]
general-log-file = /path/to/mysql/dir/mysql-general.log


Then, configure your syslog/syslog-ng to read the FIFO pipe and do with it as it will.. 

In my case, I pipe it across net to a centralized server with only the error logs and slow query logs however.

In situations where you want to also keep local copy, just set it to output to table and file as described above.