Programmatically add menu Item in WordPress

functions.php

<?php

add_filter( 'wp_nav_menu_items', 'add_logout_link', 10, 2);

/**
 * Add a login link to the members navigation
 */
function add_logout_link( $items, $args )
{
    if($args->theme_location == 'site_navigation')
    {
        if(is_user_logged_in())
        {
            $items .= '<li><a href="'. wp_logout_url() .'">Log Out</a></li>';
        } else {
            $items .= '<li><a href="'. wp_login_url() .'">Log In</a></li>';
        }
    }

    return $items;
}