How do I capture the log output from Airport Utility?
You're probably way over this, by now, but I just set up log capture from my 5th gen Airport Extreme to my Mac Mini (running Mavericks), so I'll chime in for posterity.


Syslog Destination Address does expect an IP address. The computer at the end of that ip address has to be configured to be a log server (and OSX can be configured for this... see below).
I've only just started forwarding the logs, but I've looked at the logs before.
AFAIK, apart from more directly viewing/exporting a recent sample of the logs through the "Logs &amp; Statistics" button on the advanced tab, there isn't another way to access them.
Apple removed log access (along with many other useful feature) from the Airport Utility 6 because...I dunno...they're hateful succubi seeking to relieve everyone of both money and control. You'll have to stick with 5.6.1 as long as you can.


Configuring OSX (Mavericks) to record the logs

As for configuring my Mavericks Mini to be a log server, I followed the instructions here (which I think is the same blog post that @drevicko mentioned, only in a different location).

To briefly summarize, you need to add the following to the Sockets section of /System/Library/LaunchDaemons/com.apple.syslogd.plist :


&lt;key&gt;NetworkListener&lt;/key&gt;
&lt;dict&gt;
    &lt;key&gt;SockServiceName&lt;/key&gt;
    &lt;string&gt;syslog&lt;/string&gt;
    &lt;key&gt;SockType&lt;/key&gt;
    &lt;string&gt;dgram&lt;/string&gt;
&lt;/dict&gt;


Editing the file requires that you convert it from binary to text and then back to binary once you are done.  The blog post has instructions to do this using plutil -convert

After the changes have been made, you need to restart the syslogd Daemon (instructions also in the blog post):

$ sudo launchctl unload /System/Library/LaunchDaemons/com.apple.syslogd.plist
$ sudo launchctl load /System/Library/LaunchDaemons/com.apple.syslogd.plist


Notes


When you configure the airport to send logs to a particular IP in your network, you should keep in mind that it may very well be a dynamic IP. In my case, the Airport could (and would) one day decide that my Mac Mini should be 10.0.1.5 instead of 10.0.1.2. Fortunately, you can create a DHCP reservation, in the Airport, to always give a particular machine a particular address.
Once set up to receive the airport logs, you can view them using the console application on OSX. They'll be mixed in with everything else (in the "All Messages" view), but you can find them all in the "system.log" section.