Were you able to resolve the issue? No wonder no PHP errors logged, the 500 error comes from the web server. Check its configuration. A simple double check is to put a test.html file in the root of the Moodle code directory and visit it in the browser. If successful put the usual phpinfo.php (single liner) there and visit it.
Either way, plan your upgrade path, Moodle 4.0 is not supported for one year now. The chart http://www.syndrega.ch/blog/#php-and-dbms-compatibility-of-major-moodle-releases shows your options. If I were you, after updating Moodle 3.8 to its final, I would upgrade it to 3.9 LTS. From then on you can march from LTS to LTS (4.1, 4.5).
P.S. This seems like an upgrading issue. Better suited for the Installing and Upgrading help forum.