NGinx - Count requests for a particular URL pattern
You can use the post_action directive to trigger a sub_request after the main request is complete.
Useful for the sort of logging you have in mind.
** OCT 2016 UPDATE **
The post_action directive has been removed from the Nginx documentation and while it still appears to work, usage is inadvisable. Caveat Emptor!
** JAN 2020 UPDATE **
The Mirror Module, introduced in Nginx 1.13.4, essentially replicates the post_action directive.