I installed nginx as root, now I can't get permissions right to run as another user

As the title says, I have a server where nginx was originally installed by the root user. I now want to run nginx as a non-root user, a new user that I added to both the sudo the www-data groups.

When I try to run nginx from this new user, I get the following message:

nginx: [alert] could not open error log file: open() "/var/log/nginx/error.log" failed (13: Permission denied)
2016/08/09 22:07:52 [warn] 17609#0: the "user" directive makes sense only if the master process runs with super-user privileges, ignored in /etc/nginx/nginx.conf:1
2016/08/09 22:07:52 [emerg] 17609#0: open() "/var/log/nginx/access.log" failed (13: Permission denied)

I’m confused why this is an issue—I changed the owner and group of /var/log/nginx/ to www-data. The nginx master process user is root, which I believe is correct, and the sub-processes are running (or trying to run) as www-data. The file permissions for error.log and 'access.log` are 755. What am I missing?