You might want to take a closer look at the JQuery docs for ajax requests, so you can use a secure http connection for logging. This javascript code basically describes a function that sends the errors in text-format to your server-side script. This script can in turn write the error description to a file on the server. I'd recommend using a DB instead; That way you can easily write a web-client that displays all reported errors (and filters and the other good stuff).
You can extract the origin url from the referer [sic] field in the ajax http get-request on the server.
(function () { // function operator, in case console doesn't exist
    !console ?
        (console = {}) : console;
    !console.log ?
        (console.log = function () { }) : console.log;
    !console.info ?
        (console.info = console.log) : console.info;
    !console.error ?
        (console.error = console.log) : console.error;
}());
// Uses JQuery
function reportError (errDesc) {
    var path = "www.getlog.com/mylogs.php";
    $.ajax({
        url: path,
        type: "GET",
        async: true,
        cache: false,
        contentType: "application/x-www-form-urlencoded; charset=UTF-8",
        crossDomain: true,
        data: errDesc,
        dataType: "jsonp",
        error: function (req, type, errObj) {
            console.error("Reporting error failed: " + type + "\nAt url: " + path + "\n" + errObj);
        // In case you need to debug the error reporting function
        },
        succes: function (res) {
            console.info("Reported error to server:\nRequest:" + errDesc + "\nResponse: " + res);
        // extra error logging facility on client-side, invisible to most users
        },
        global: false // prevent triggering global ajax event handlers
    });
    return errDesc; // in case you want to reuse the errDesc
}
Code has been validated with jshint. Please let me know if there are still issues, because I didn't take the time to completely replicate your setup (setting up 2 different domains etc.)
Addendum: Some useful reading if you're having issues with cross-domain messaging, JSON is not a subset of javascript, Cross-origin resource sharing, JSONP.