Wordpress - What log files does a WordPress installation write to?

Where (or if) errors are logged is dependent on your php configuration. You can control some of this via your WordPress wp-config.php file. See Configure Error Logging in Editing wp-config.php.

If you have access to it on that system (unlikely on a shared host) have a look in the messages log. I've found most failures will log something in there. You might also check the web server logs. As far as I'm aware Wordpress doesn't generate it's own logs.