In my case php-fpm outputs 500 error without any logging because of missing php-mysql module. I moved joomla installation to another server and forgot about it. So apt-get install php-mysql
and service restart solved it.
I started with trying to fix broken logging without success. Finally with strace
i found fail message after db-related system calls. Though my case is not directly related to op's question, I hope it could be useful.