isOpen at home?

For the pages in my menu i can use:

        <?php foreach ($site->children()->listed() as $item): ?>
        <li><a class="hover:text-indigo-300 <?php e($item->isOpen(), 'text-indigo-400 ') ?>" href="<?= $item->url() ?>/"><?= $item->title()->esc() ?></a></li>
        <?php endforeach ?>

How can i know that i am on my home/index to color the text of the logo-text also?

HOME ====== Menu 1, Menu 2, Menu 3

In the same way, but not in that list, because your logo is not in the list. So you need to apply that logic for the home page link.

Γ„hem, yes, shure, but what is the var to check?

site()->homePage()->isOpen()

mind the capital P.