You can leverage the Diagnostic Log extension on Azure Web Apps. Login your web app in Azure portal(https://ms.portal.azure.com/). Click on Diagnostic logs in settings option and Turn On Application logging in Diagnostic Logs Tab.

And then you can login the Kudu console site of your web app(https://<Your_Webapp_name>.scm.azurewebsites.net/DebugConsole) and browse to your Application folder(D:\home\LogFiles\Application folder).

Otherwise, you can click the Diagnostic dump on the top nav bar of the kudu console site to download the log files.

On the other hand, you can use the Log stream tool on the Azure portal(https://ms.portal.azure.com/) for a real time debugging online.

At last, you can integrate a 3rd part node.js log modules, such like winston into your application. And catch the logs into your own log file.
Any further concern, please feel free to let me know.