Log profile 2 user info a separate log file
The profiles2 docs state:

Thanks to the foundational entity API the module integrates well with Rules as well as Features.

Install the rules module (http://drupal.org/project/rules) and you should be able to access the profile data after he user login event. To log to the system watchdog with rules you need this patch: http://drupal.org/node/1349882#comment-6190164

This does not write to a separate file, but it would be trivial to extract your entries from the watchdog db table using phpMyAdmin.

And finally it does not require any coding.