I put together http://finderr.org and use that as a search service exactly because of this. I don't keep IPs in the logs of the server.
I'm not sure how to go about the log configuration. If the users prefer it I can just disable logging altogether, although I'd prefer keeping some information (not IPs) at least for a single day so I can get basic usage statistics. I promise not to do anything evil with the little information I may get in any case.
Also: don't only promise! Put privacy page right there and explain what you explained here (maybe even in greater details, people love transparency), so every user we'll be able to obtain same information.
I agree that an extension may be better, although this may come handy when you're not using your computer or main browser.