Change a url / link if a user is logged in?

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.

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