Microsoft ASP.NET WebHooks uses logging as a way of reporting issues and problems. By default logs are written using System.Diagnostics.Trace where they can be manged using Trace Listeners like any other log stream.
When deploying your Web Application as an Azure Web App, the logs are automatically picked up and can be managed together with any other System.Diagnostics.Trace logging. For details, please see Enable diagnostics logging for web apps in Azure App Service
In addition, logs can be obtained straight from inside Visual Studio as described in Troubleshoot a web app in Azure App Service using Visual Studio.
Instead of writing logs to System.Diagnostics.Trace, it is possible to provide an alternate logging implementation that can log directly to a log manager such as Log4Net and NLog. Simply provide an implementation of ILogger and register it with a dependency injection engine of your choice and it will get picked up by Microsoft ASP.NET WebHooks. Please see Dependency Injection in ASP.NET Web API 2 for details.