When I add the package ServiceStack.Logging.Serilog to my project
I get this exception:
Package ServiceStack.Logging.Serilog 4.5.8 is not compatible with netcoreapp1.1 (.NETCoreApp,Version=v1.1)
Any particular reason why this package is not support on netcoreapp 1.1?
ServiceStack logging packages are published only for .NET 4.5, they are not needed for enable logging on .NET Core. You can configure logging in standard ASP.NET Core way as described here and your services will use it. For particular Serilog configuration you can look at this and this posts.