To start off, make sure your log files are formatted consistently before parsing them. If you just want to display the logs without parsing, then that step may not be necessary. However, if there is a lot of parsing involved, it's a good idea to store the parsed data as log_entries in your database for quick access. Once that's done, all you need to do is send the data from your server to your web browser. You can create a simple HTML page that refreshes the log view either at set intervals or when a button is clicked.
When designing the log view, you'll probably use a server-side language unless you store your log files in a static directory served by your web server. PHP, Python, Ruby, and Perl are common choices for this task, with Perl being particularly great at reading logs. It's recommended to serialize the data sent to the browser as JSON or XML. JSON is popular and easy to work with in JavaScript – simply parse the JSON string into a JavaScript object using JSON.parse(str).
For presenting the log data in an appealing way, consider using a UI library. While creating custom widgets can be beneficial for simple applications like this one, it can also be time-consuming especially for beginners. Some popular UI libraries include jQuery UI, Sencha (formerly ExtJS), and Closure Library's UI package (goog.ui).