When his arm gets trapped under a giant rock in the Grand Canyon, a man must decide how much he values his own life and how far he will go to preserve it.

127 Hours (2010). The movie was based upon a real life incident. Noting that in the logline strengthens the hook of the logline.

The hook of the logline is that it's a true story of a man who faces an awful dilemma: he must cut off his arm -- or die. It's a do or die plot predicament -- and that's what the logline should focus on. That predicament was the story hook that attracted the interest of acclaimed director Danny Boyle.

So, maybe:
The true life story of a mountain climber who, when his hand gets trapped under a boulder in a remote canyon, must cut it off or die.

(27 words versus 33 words for the logline posted. With loglines, less is more.)