This tutorial describes how you can enable MySQL to log every query to a file. All of this is possible without the need to restart MySQL.
Create query log file
touch /var/log/mysql_queries.txt
Set file permissions so that MySQL server can write to the file
chown mysql:mysql /var/log/mysql_queries.txt
Enable query logging
Login to the MySQL server and the run the following queries:
SET global log_output = 'FILE';
SET global general_log_file='/var/log/mysql_queries.txt';
SET global general_log = 1;Disable query logging
Login to the MySQL server and the run the following query:
SET global general_log = 0;