iPhone 5 battery draining very quickly
I wouldn't necessarily take Apple's word on how much of use you can get out of your iPhone given it's mostly marketing and reproduced in specific lab conditions. Here are some real world numbers from AnandTech's iPhone 5 review. The differences can be quite significant for instance, 4.55 hours on 3G and 8.19 on LTE.  

Note that this is presumably for a brand new iPhone 5; if you have one that is more than a year old, it would probably be around 80-90% of it's maximum battery capacity.

Lastly, here are some ideas on maximising your iPhone's battery life:


Disable Location and Background App Refresh for Facebook
Disable Background App Refresh for Apps You Don't Care About
Stop Quitting Your Apps in Multitasking
Disable Push Email Temporarily (especially if you are using Exchange)
Disable Push Notifications for Apps That Annoy You


It may also be worth doing the test mentioned at the top of that blog post i.e. 


  So here's the test: write down your usage and standby time, press the
  sleep/wake button (or lock button, as some call it) to put the device
  to sleep, and set the device down for five minutes. When you come
  back, take note of the change in time. If your device is sleeping
  properly, then the Standby time should have increased by five minutes
  and your Usage time by &lt;1 minute . If your Usage time rises by more
  than one minute, you have a drain problem. Something is keeping your
  device from sleeping properly, significantly shortening the time it
  will last.


From my own experience, I've noticed that apps that utilise background app refresh in concert with location services tend to be pretty expensive in terms of battery usage. Dropbox and Downcast use these to upload photos and download podcasts in the background respectively - location services appear to be turned on all the time when these Dropbox/Downcast features are turned on. I turned off background app refresh and location services for both of these apps and have noticed quite significant improved battery life.