I currently cannot access
http://example.com/admin/reports/dblog as I cannot authenticate as an admin as my page is getting a 500 error.
I wonder if there is an actual location on my production server where Drupal errors are logged.
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If you have drush set up, you can use
drush watchdog-list to look at the items logged to watchdog.
If you don't have access to drush, but your DB is up and running fine, you can always query the watchdog table with
Select * from watchdog;
From personal experience, you might want to check your apache error logs as well to see if those hold anything that can help.