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You could sniff the other person's traffic using a man-in-the-middle attack. But I don't know why you would want that.
 
You could install a server and run something like WebSense.

The Airport doesn't support logging.
 
I'm sure I could do a lot of things, but can I use my airport to log their websites?

Well, if you are connected to the network via AirPort, I don't think there is another possibility than using your AirPort.
 
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