I put my phone in DFU mode and did a restore and it works fine now. The battery lasts at least as long as my iPhone 4. So I did the same thing to my Wife and my Mother's iPhone 5's. They both are working fine now with normal battery life. As it said on another thread it appears to be a bad image from the factory. I did exactly what t.katz said below. Has fixed three phones so far.
t.katz
Hey Blackout,
What I did was forced a backup through iTunes (connected via the usb cable), and then put the phone into DFU mode and did a restore. Then I went through the usual iTunes process to restore my backup to the phone. Doing the restore via iCloud should work as well, I just find it much faster to restore via the iTunes backup then over WiFi. I like to keep local backups in iTunes anyway..
I don't know if entering DFU mode vs just doing a straight restore actually makes a difference, its just something I got used to do doing. Whenever it seemed like my phones battery was draining fast, or things were acting weird I would always DFU restore and that would seem to make the issues go away. It could simply be psychological giving the feeling of doing a "format + install" vs simply a "reinstall" type thing, I have no idea at this point 😝
As for entering DFU mode, I'm not sure if thats something Apple likes discussed on these forums? Just incase, if you do a google search... oh, say perhaps "iphone restore dfu", you might just find your answer in the first result (iClarified).
Hope this helps.