Capturing javascript console.log?

Simple:

function yourCustomLog(msg) {
  //send msg via AJAX
}

window.console.log = yourCustomLog;

You might want to override the whole console object to capture console.info, console.warn and such:

window.console = {
  log : function(msg) {...},
  info : function(msg) {...},
  warn : function(msg) {...},
  //...
}

You can hijack JavaScript functions in the following manner:

(function(){
    var oldLog = console.log;
    console.log = function (message) {
        // DO MESSAGE HERE.
        oldLog.apply(console, arguments);
    };
})();

1. Line 1 wraps your function in a closure so no other functions have direct access to oldLog (for maintainability reasons).
2. Line 2 captures the original method.
3. Line 3 creates a new function.
4. Line 4 is where you send message to your server.
5. Line 5 is invokes the original method as it would have been handled originally.

apply is used so we can invoke it on console using the original arguments. Simply calling oldLog(message) would fail because log depends on its association with console.

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Update Per zzzzBov's comment below, in IE9 console.log isn't actually a function so oldLog.apply would fail. See console.log.apply not working in IE9 for more details.