Well to do that, you'd have to have it writing to the admin logs.
If you use the above example access.log - then there's no script which reads that. The script you are talking about reads the admin logs.
So you could have your thing write to the admin log instead then you could read it with the admin log stuff, or you you could duplicate the script which reads the admin log - and tweak the filename that it reads, and add a link for it in the admin menu.
That's what I did. I can't post it as it's pretty much almost an identical copy of the script which reads the admin log with just the filename changed (and it's a copyrighted script), but that gives you the idea.