Hey my iphone 5 had battery issues too, I had it jailbroken so that was the reason lol, the reason your battery is draining could be any number of things so you'll have to kind of start broad and then get more specific
First - it could totally be a failing battery. It happens... I think they can diagnose that over chat, phone, or at the genius bar... If it says the battery needs service *boom* new battery! but it's most likely software or settings...
Is this a new issue or was it always like that? If it was always like that then you're looking at either a corrupted backup or some software draining it. If it's new you're looking at maintanance or settings. How often do you turn off the phone? Iphone 5 wants to be powered off every other day. Just give it a minute to close all the processes and clear the memory. Is everything up to date? ios? all the apps?
I know when the google app updated with Google Now it had a new service that used location services all the time, did you check all the settings at: http://www.apple.com/batteries/iphone
I live in new york and can be on my phone underground for 1 hour. If wifi/cell data is enabled the phone will consistantly search for a network. Try turning on airplane mode if you're in an area with no reception for long periods of time.
When you started up the phone did you use a backup to get everything on it? It's possible that copy isn't great and the solution might be restoring to factory settings and starting the phone as new. Like a backup from a 3G iphone to an iphone 4 then an iphone 5, it could be like a messy xerox. See how old your text messages are, that might be the easiest way to tll how old the data on the device is.
If you restored and setup as new, and optimized the settings, and the battery doesnt need service then it may be something trickier like photostream or syncing services running, facebook, email services... but start broad, turning it off regularly might fix it... good luck!