When waking from sleep, computer shuts off

I was given this MacBook for free so before I invest an money into it, I want to pinpoint the cause of the problem.

When you shut down the MacBook and press the power button to turn it on, it comes on for a second or two (the startup chime to starts to play) and then the power shuts off.

Also, when you close the lid and let it sleep, opening the lid again will wake it up for a second or two and the power shuts off.

To get it to work, I have to hold the power button wait for the status light to blink rapidly and then it will boot up.

Is there a log file I can look at to see if some voltage threshold is being breached and it is shutting down to protect it? Other than this problem, the laptop functions perfectly.