Susan,
Having the computer shut down at the manufacturer's logo or during P.O.S.T., is a prime symptom of overheating.
The first thing you should do is get a can of compressed air and blow out the vents at the side of the laptop.
The second thing you should do is confirm that the fan inside (if there is a fan) is actually rotating to circulate air away from the processor.
There are other things that COULD cause this symptom, but this is the most common for most/all laptops.