You have one pair of Nginx logs server-wide and another for each site, unless you omitted them. The error logs do just that, logging any errors. The access logs show what web elements have been resolved by IP address and when.
Those who followed Serve Multiple Sites & Blogs with Virtual Hosts mapped out the location of the logs in each site or blogs virtual host file or, if you bundled each site's configuration in your nginx.conf file, the locations to the files are given there.
Here's a reminder anyhow.
Nginx Web Server Error & Access Log Locations
To list them:-
Or to open them:-
.. substituting error.log for access.log as appropriate.
Or if you installed Nginx using the package manager, rather than compiling from source as is recommended in the VPS Bible, they're here:-
Web Site Error & Access Log Locations
And the error page:-
My Error or Access Log is a Blank File/Won't Open.
In the case of the error log, hopefully it is indeed empty. With the access log, maybe it's just too darn big and is taking a long time to open. If you really want to look at it, be patient. Else delete it and add a new, empty one:-
.. saving the empty file.