Maybe there’s something wrong with your iPhone 5’s battery after all. Apple has found that a “very small percentage” of iPhone 5 units (the 5, not the 5s, and not the 5c) “may suddenly experience shorter battery life or need to be charged more frequently.” If that sounds like your iPhone, check out Apple’s serial number checker on their iPhone 5 Battery Replacement Program page and see if your phone qualifies. Apple will replace the battery at no charge.
Note: this program went into effect on August 22nd, 2014 in the United States and China. It will be active worldwide starting August 29th, 2014.
If your iPhone doesn’t qualify for the battery replacement program, and even if it does, check out my post on stretching your iPhone’s battery life for some tips.