Can I extend the console object (for rerouting the logging) in javascript?

You Can Also add log Time in This Way :

added Momentjs or use New Date() instead of moment.

var oldConsole = console.log;
console.log = function(){
    var timestamp = "[" + moment().format("YYYY-MM-DD HH:mm:ss:SSS") + "] ";
    Array.prototype.unshift.call(arguments, timestamp);
    oldConsole.apply(this, arguments);
};

Try following:

(function() {
    var exLog = console.log;
    console.log = function(msg) {
        exLog.apply(this, arguments);
        alert(msg);
    }
})()

It's really the same solution some others have given, but I believe this is the most elegant and least hacky way to accomplish this. The spread syntax (...args) makes sure not a single argument is lost.

var _console={...console}

console.log = function(...args) {
    var msg = {...args}[0];
    //YOUR_CODE
    _console.log(...args);
}