I do want to keep the old information. I'm also limited on how many tables I can make so I wanted to keep as much information in a single table as possible. However, I was able to find what I was looking for.
My new code looks like so:
<?php
// New entry to be added into log
$logged_activity = time() . ": User Login<br /> \n";
// Table update user's log
$query = "UPDATE users SET user_log=CONCAT('{$logged_activity}', user_log) WHERE id=1 LIMIT 1";
mysql_query($query) or die("Update Failure: " . mysql_error());
?>
This almost works perfectly... almost. If the field is NULL to begin with, it will not add the new information. But I can circumvent that by making the field blank or contain a single space during user creation.