Singleton Pattern in JavaScript Going over the though process of implementing the same, with a minimal to an optimized route

singleton.js

/*************************************************
   *     SINGLETON PATTERN IMPLEMENTATION          *
   *************************************************/

  //since there are no classes in javascript, every object is technically a singleton
  //if you don't inherit from it or copy from it.
  var single = {};
  //Singleton Implementations
  //Declaring as a Global Object...you are being judged!
  
  
  var Logger = function() {
    //global_log is/will be defined in GLOBAL scope here
    if(typeof global_log === 'undefined'){
      global_log = this;
    }
    return global_log;
  };
  
  
  //the below 'fix' solves the GLOABL variable problem but
  //the log_instance is publicly available and thus can be 
  //tampered with.
  function Logger() {
    if(typeof Logger.log_instance === 'undefined'){
      Logger.log_instance = this;
    }
  
  
    return Logger.log_instance;
   };
  
  
  //the correct way to do it to give it a closure!


  function logFactory() {
    var log_instance; //private instance
    var _initLog = function() { //private init method
      log_instance = 'initialized';
      console.log("logger initialized!")
    }
    return {
      getLog : function(){ //the 'privileged' method 
        if(typeof log_instance === 'undefined'){
          _initLog();
        }
        return log_instance;
      }
    };
  }
  
  /***** TEST CODE ************************************************
  //using the Logger singleton
  var logger = logFactory();//did i just gave LogFactory a closure?
  //create an instance of the logger
  var a = logger.getLog();
  //do some work
  //get another instance of the logger
  var b = logger.getLog();
  
  
  //check if the two logger instances are same?
  console.log(a === b); //true
  *******************************************************************/