Actually both theme I’ve tested polylang with have automatically this behaviour, landing page switch to browser’s default if available, and while browsing if i change language on a page and click the logo, I reach the landing page in the same language i was one click before…
I have for landing page a ‘static page’, this page#1 has a translation (page#2), polylang seems to point automatically to the correct one according to current language.
You would have to replace the link over your logo (in header.php probably) with a standard
<a href="<?php bloginfo( 'url' ); ?>/" title="<?php bloginfo( 'name' ); ?>">
But it depends on how you’ve set wordpress landing page too I guess…
By the way you should give all your settings as described in the top post of support forum too, that would help and it is anyway described as mandatory prior posting …