tarppa

Write all login attempts where both username and password are supplied to a log file. The logfile is relative 
to error.log and is called ‘login.log’, the format : “TIMESTAMP USERNAME SUCCESS/FAILURE”.
In practice this plugin is a rewritten version of the wp_authenticate_username_password function from user.php and replaces 
it in the login process.

WP Login Logging

Write all login attempts to a logfile by replacing default wp_authenticate_username_password functionality

PhilHilton

This plugin is not affiliated with WP Forms, makers of the WP Mail SMTP plugin. Please do not bother them with support requests. It is a 3rd party solution to the lack of logging in the free version of their product.
Please bear in mind that future updates to WP Mail SMTP may break this plugin’s ability to log messages, and it may take some time for an update to be released. If they go out of their way to remove the WordPress Action Hook it depends on then an update may not be possible at all. Please use this plugin at your own risk.
This plugin has only been tested with the SMTP configuration, and not any of the other implementations provided by WP Mail SMTP. If you use it with another configuration I would be interested in hearing about you experience so that I can either confirm here that it works or make changes so that it will.

Logger for WP Mail SMTP (3rd Party)

This plugin is not affiliated with WP Forms, makers of the WP Mail SMTP plugin. Please do not bother them with support requests.

WP Login Logging vs Logger for WP Mail SMTP (3rd Party)