If this is for debugging and you have control over the page then yes. You will have to allow ActiveX controls to run. If possible add the site to "Trusted Sites" and permit the ActveX settings at the top of the security settings. You can remove the site later if you wish.
Use the following function:
var FILE_PATH = "C:\log.txt"; //Change to wherever
var ForAppending = 8;
var oFSO = new ActiveXObject("Scripting.FileSystemObject");
var oTS = oFSO.openTextFile(FILE_PATH, ForAppending, true);
oTS = null;
oFSO = null;
and then replace all alert calls with Alert (Capital A).
A simpler solution if you don't need a permanent record is to insert a div in the page with an id of "divMessage" and use:
document.getElementById("divMessage").innerText = Text;
Another thing is to press alt+SPACEBAR when the large alert box is open and move the alert up the page.