Using jQuery to delete data stored in wp_options

I wonder if anyone could advise me further on my problem. Part of my plugin stores log files for debugging purposes. I have successfully displayed them in a (div#log) in my admin page using jquery and wp_localise_script. I have a button to delete these logs but I am unsure how to process this. I have a feeling that ajax might come in handy here but not sure where to start.

How to implement AJAX post navigation into WordPress?

I am developing a wordpress page template exactly like this.
I have completed the design and have loaded the template once user goes to this page which using my template. I want to all posts in more article list open in this template without refreshing the page. but when i click on any post in the list it opens in a new page which is off course different from my page template. Please guide me how i can open the posts in the “more article” list open in my template.
Here is my code i have till now.