How to check mysql password change and other events logs
by default MySQL community version not support audit

You have only one build in way - enable general log for log all activity on MySQL server 

alternatives for enterprise version - https://www.percona.com/doc/percona-server/5.5/management/audit_log_plugin.html

MariaDB audit plugin You can install and on Linux and on Windows server - https://mariadb.com/kb/en/mariadb/about-the-mariadb-audit-plugin/