If you use basic authentication, you really can't log out unless you completely quit your browser.

This type of authentication requires the browser to identify itself for each request. Since it doesn't want to "bother" the user every time, the browser caches username and password and automatically sends it along with each request.

From your question, I'm guessing you have users that have roles. If a user does not have a role, you want to allow a different user to log in. Since you can't do that, the only thing you can try is "invalidate" the whole user account as soon as that user is not in a role you require. Say you start with first.html and log in as user bob with password bob, the authentication mechanism will look up bob in the database, check the password and allow access if bob has the role required for first.html. Now suppose you go to second.html from there and that only user joe has a role required for that. You first get a request for second.html with bob/bob. You see it's bob, you see bob doesn't have the right role, so you simply refuse bob access here by just pretending he supplied the wrong password. The user then will have the opportunity to switch to joe for this page.