How to Add a Login/Logout Link to Your WordPress Menu

add_filter('wp_nav_menu_items', 'add_login_logout_link', 10, 2);
function add_login_logout_link($items, $args) {
        ob_start();
        wp_loginout('index.php');
        $loginoutlink = ob_get_contents();
        ob_end_clean();
        $items .= '<li>'. $loginoutlink .'</li>';
    return $items;
}