BuddyPress – How to add logout in nav menu

Google is your friend. Here is a snippit I found I didn’t test it but seems logical.

 // functions.php
function add_login_logout_link($items, $args)
{
  if(is_user_logged_in())
  {
    $newitems="<li><a title="Logout" href="". wp_logout_url('index.php') .'">Logout</a></li>';
    $newitems .= $items;
  }
  else
  {
    $newitems="<li><a title="Login" href="". wp_login_url('index.php') .'">Login</a></li>';
    $newitems .= $items;
  }
  return $newitems;
}
add_filter('wp_nav_menu_items', 'add_login_logout_link', 10, 2);

Source