Q:

Change admin logout URL

My site is using a short vanity URL.  This is working across all of the custom theme pages.  The only problem is when I logout in the admin it redirects to the long URL, which then 404s.  
My General Settings (which I can't change) are:
WordPress address (URL): The long URL
Site address (URL): The short URL
Looks like the logout link is generated from wp-includes/general-template.php, but I hate to edit core, non-theme, WordPress files.  
Any ideas how to solve this logout 404 problem is greatly appreciated.  

A:

The quickest way to do this is through an Apache rewrite via mod_rewrite.  You'll also have to tell WordPress where to points its login links using the login_url and logout_url filters. 
  return apply_filters('login_url', $login_url, $redirect);
  return apply_filters('logout_url', $logout_url, $redirect);

