Change the line with the URL:
<a href="https://wordpress.stackexchange.com/questions/70644/<?php echo get_option("home'); ?>/"><div id="logo">
… and check if the user is logged in:
<a href="https://wordpress.stackexchange.com/questions/70644/<?php
if ( is_user_logged_in() )
{
echo admin_url("/themes.php?page=wptuts-settings' );
}
else
{
echo home_url( "https://wordpress.stackexchange.com/" );
}
?>/"><div id="logo">
And echo get_option('home');
is not correct.
See wp-includes/link-template.php:home_url()
to understand why the real home URL is a little bit more complex.