I did that, and the apple logo is up... trouble is, it's not going away.
ETA: Huzzah! after about 5 minutes the main screen came back! Thanks so much! I was getting so scared -- my daughter loves this phone!
Ah, I can tell you what probably happened, then. Something caused mild corruption of the HFS partition in flash memory; that doesn't happen often, and when it does, the iPhone is capable of recovering from this. (This is true of most modern technologies, and they'll usually recover during boot. It's why you might see the blue CHKDSK screen on Windows, or see the grey Apple loading screen stay up a long time with the little activity spinner on a Mac, or see the fsck routine in a text-based Linux boot...)
This is why 'try rebooting' is actually not bad advice for almost any technical issue; most systems do their automatic self-check and repair type routines on boot. Even modern cell phones.
On the iPhone the Apple logo stays up during the entire boot sequence -- basically until Springboard starts (Springboard being the software that provides the iPhone's 'Home screen'), so if it's doing that type of repair you'll just see the Apple sitting there for a very long time.