Don't think so out of the box (correct me if I'm wrong...). 

The sweetest solution I think is to inherit the membership provider and add this functionality to validate validate user method. Change web.config under membership to use your improved provider...

Another way is of course to make a custom login page. Also pretty easy to do and add logging there...

Thank you Daniel, that's what I suspected.

I think I will create a custom login page and hide the util/login.aspx-page so I have full control of what happens in the login functionality.

If you just need to log failures, overriding membership would be best choice. If you need to customize look and feel of the login page, etc -> then of course new page is needed.