Then why not just delete all cookies if you leave FR and go to FB? These are free and it only takes seconds. I run them both several times a day:
http://www.downloadatoz.org/ccleaner/
http://www.stevengould.org/index.php?option=com_content&task=view&id=15&Itemid=69
Because cookies are handy. They let me click on FR over and over and I never have to remember my password.