The get_template_directory() and get_template_directory_uri() functions don’t actually echo anything at all. They will make no output at all unless you echo them.
So your code that looks like this:
<img class="frontpageimg" src="https://wordpress.stackexchange.com/questions/119910/<?php get_template_directory(); ?>/images/slogan-2.png">
Should actually cause this output:
<img class="frontpageimg" src="https://wordpress.stackexchange.com/images/slogan-2.png">
Which may not be what you’re expecting. Examine the source of the page. And add that echo, it’s not optional. 🙂
Also, if the theme may have screwed with relative urls and rewrites and such, as Chip suggests, then your browser is caching. Browser caching is notoriously difficult to debug. I like to use Chrome and the inspector tool, because when you have it open, right-clicking on the reload button gives you a “Empty Cache and Hard Reload” which does more what you expect it to do as far as showing the real truth of the matter.