On the top left of your news feed, you'll see a "list" button that expands into a detailed menu. At the very bottom, there is a set of account settings; the last of which allows you to log out of your account. This is version 6.4, on iOS7.
If it isn't logging you out, you have possibly connected your phone to your Facebook account. You can change this in the Settings app; logging out of both of these will solve your problem.
Good luck!