What is the Behavior of a Singleton in ASP.NET Web Services?

I want to create a logging component to log errors in my Web services layer to a database through a proprietary API. The Web services layer can be distributed across multiple application servers. I believe the logging component should be a Singleton. What is likely to happen when I get the instance of the Singleton? Do I get a separate logging Singleton instance on each application server? My Web services layer is written in ASP.NET. My logging component is loaded dynamically through System.ComponentModel (a.k.a. MEF).