It only has a 3X optical zoom but can also do movies and is in your price range. Its key feature is a 'workable' 1600 ISO capability which for a compact camera is really your only hope of getting any kind of consistent quality for indoor snapshots, family photos, and moving dogs. Without this, as a general rule, compact digital cameras suck for indoor/lowlight photography unless you're just a few feet from your subject. By their very nature their sensor has neither the light gathering ability nor is the built-in flash powerful enough, so that sometimes you get great shots but more than half the time the subject is not quite in focus and/or not well lit. Anyway, I don't own this camera but I've heard some good things about it:
http://www.dpreview.com/reviews/fujifilmf10zoom/BTW, I use an Olympus C-7000 which has the limitations I describe (ISO only up to 400) but when on a tripod or outdoors the pictures it takes are stunning ('SLR-like' I've heard others say).