There is no log file. Each node.js "app" is a separate entity. By default it will log errors to STDERR and output to STDOUT. You can change that when you run it from your shell to log to a file instead.
node my_app.js > my_app_log.log 2> my_app_err.log
Alternatively (recommended), you can add logging inside your application either manually or with one of the many log libraries: