protected string Source
  {
    get
    {
      StackTrace st = new StackTrace(true);
      MethodBase mb = st.GetFrame(1).GetMethod(); 
      return mb.ReflectedType.Namespace + "." + mb.ReflectedType.Name + "." + mb.Name; 
    }
  }

//Use like so: 
  public void DoChildStuff()
  {            
    Log(Source, "Doing Child Stuff");
  }

GetFrame(1) to avoid the last method being this property accessor.  
Note that in a child class inherited member, the Class name will be the name of the type containing the implementation.

Get the Namespace, Class Name, and Method Name of the Currently Executing Method Via Reflection

C#