There is no PHP function alert, that’s why your function throws an error and doesn’t proceed with the rest of your code. You have to return something from the LogHit_callback function using return (or echo it with echo) and then alert it with JavaScript, that is add following code to your post function:
jQuery.post( 'http://www.test.com/wp-admin/admin-ajax.php', data, function( r ) {
alert( r );
});
I believe that this is the correct syntax for post(). Remember that you can always debug AJAX requests with browser’s developer tools. You will then know exactly what’s going on.