How do I do logging in C# without using 3rd party libraries?

public void Logger(string lines)
{
  //Write the string to a file.append mode is enabled so that the log
  //lines get appended to  test.txt than wiping content and writing the log

  using(System.IO.StreamWriter file = new System.IO.StreamWriter("c:\\test.txt", true))
  {
    file.WriteLine(lines);
  }
}

For more information MSDN

I would rather not use any outside frameworks like log4j.net.

Why? Log4net would probably address most of your requirements. For example check this class: RollingFileAppender.

Log4net is well documented and there are thousand of resources and use cases on the web.

You can write directly to an event log. Check the following links:
http://support.microsoft.com/kb/307024
http://msdn.microsoft.com/en-us/library/system.diagnostics.eventlog.aspx

And here's the sample from MSDN:

using System;
using System.Diagnostics;
using System.Threading;

class MySample{

    public static void Main(){

        // Create the source, if it does not already exist.
        if(!EventLog.SourceExists("MySource"))
        {
             //An event log source should not be created and immediately used.
             //There is a latency time to enable the source, it should be created
             //prior to executing the application that uses the source.
             //Execute this sample a second time to use the new source.
            EventLog.CreateEventSource("MySource", "MyNewLog");
            Console.WriteLine("CreatedEventSource");
            Console.WriteLine("Exiting, execute the application a second time to use the source.");
            // The source is created.  Exit the application to allow it to be registered.
            return;
        }

        // Create an EventLog instance and assign its source.
        EventLog myLog = new EventLog();
        myLog.Source = "MySource";

        // Write an informational entry to the event log.    
        myLog.WriteEntry("Writing to event log.");

    }
}