1.First cross check /etc/my.cnf to make it permanent
slow_query_log = 1
slow_query_log_file = /var/log/mysql_slow_query.log
2.Permission of
mysql mysql 763482 Oct 6 11:43 /var/log/mysql_slow_query.log
3.Slow query logging without restart mysql
login to mysql as root
mysql> set global slow_query_log = 1;
mysql> set global slow_query_log_file = '/var/log/mysql_slow_query.log';
mysql> flush privileges;
Cheers!!!!!!!