Is there a way to log outbound HTTP requests that pass through my Airport Extreme?
I would suggest you to install Squid  HTTP Proxy through Brew Package manager

Squid is a HTTP Proxy and will log and cache all Browser requests passing through it.Since we are installing Squid on your Mac, it will log only the one's passing through it.

Procedure :



Install Brew using Terminal.app 
 Make sure Brew is ready to be used by used by running following command: 

$ brew doctor 

If not ready follow the instructions.

(Generally this requires you to upgrade XCode to latest version.


Install Squid

$ brew update && brew install squid 

Point your Browser  to use Proxy localhost:3128