Click here for a list of available Geocaching.com API calls. GetWptLogTypes looks like it might be what you want at first blush, but when I tried it I literally got the log types, not the logs themselves.
I don't know how Project-GC is doing it - it may be that they are scraping the Geocaching.com website, which is a violation of the TOU.
I don't know how Project-GC is doing it - it may be that they are scraping the Geocaching.com website, which is a violation of the TOU.