Hi Cat, taking photos of fast motion (karate) in low light, is really hard. Almost all point and shoot cameras have the same sized image sensor (they're small) so they all do poorly in low light.
Using zoom in low light makes things even worse. I think either of these two will do fairly well compared to other point and shoots, but don't expect anything too amazing.
There a few point and shoot cameras with larger sensors, but they don't have much zoom, tend to be advanced and pricey. You might consider something like the Nikon P7100, with a 1/1.7" sensor and 7.1x zoom.
