Since there is no mechanism defined to log the user out of a HTTP authenticated session you have to use a workaround. Most browsers forget their login data once they received a 403 response. The problem is, that the user have to return to the normal starting page manually.

Update 2017-02-22: It seems that this technique doesn't work anymore in recent Firefox and Chrome browsers.

The following code, which can be embedded into an Apache VHost declaration, uses a cookie to send a 403 response to the browser the first time he requests "/logout", and redirects it to "/" on the second time. So the user can immediately login again if he want to. 

Please note that the order of the declarations does matter! The Location-Block must be the last block which requires a user or group.

RewriteCond %{HTTP_COOKIE} ^.*logout=1.*$
RewriteRule ^/logout$ / [R,L,co=logout:0:%{HTTP_HOST}:0:/logout]                                                                                                         
RewriteRule ^/logout$ - [R,L,co=logout:1:%{HTTP_HOST}:0:/logout]
<Location /logout>
    Require user nonexistinguser
</Location>