well first of all, when I first got it I didn't charge it for a while because I didn't know I should do so and everyone tells me this is why the battery acts this way... and i don't have LTE in my country. but any way i'll give you an example - yesterday i used it only for one hour and i didn't play any games or something that drains the battery and it went to under 65% (i checked in settings how much time have passed). according to this situation and others the battery drains out in about 3-4 hours only with surfing the web and playing music. and i always close all applications.
If you're constantly closing out apps I would say that might be one reason your battery life is suffering. It might not be the direct cause and a lot of people will tell you that you should close them out if you want. However if you're trying to diagnose an issue you've got to try changing your regular routine and see if it has any affect. I've had battery issues because I'm running the beta iOS 6.1 so here is what I would tell you to try and see if your battery life gets better.
Backup your iPhone through iCloud
Hook up to iTunes and then go through each tab (music, podcast, tones, photos, etc.) except apps don't touch the apps
Each tab uncheck Sync.... (this will remove all the media from the device) once you've unchecked them all apply and do a sync
Then right click your device and choose transfer purchases
Reboot the device
Hook up to iTunes and click restore this will reload the OS
When it finishes and asks to "setup as new or restore from backup" in iTunes unplug the phone(if you did an iCloud backup)
On the phone go thorugh all the settings and login to your Apple ID and choose to restore from iCloud backup choose the last backup
Let the restore complete then reboot the device
All the media will now be back on the phone because it was restored through the iCloud backup
If you want to sync the media using iTunes then follow these steps if not then skip the following
Hook up to iTunes and again uncheck all the sync options for media except apps,do a sync
Reboot the device then hook up to iTunes and check the sync options to put the media back onto the device
Use your phone all day and when it gets low or when you're going to sleep close all the applications and plug it in. After it's plugged in power it off and let it charge until you wake up or whatever. Then use it and see if the batter life is any better.
I used to religously close out my apps until I found that my battery life sucked. Then I started to just let the OS manage the apps and the battery life got a lot better. So now what I do is if there is an app I wont use after I'm done like a FollowFriday Generator, photo app I'll kill it. If an app is misbehaving I'll close it out or if it's a huge memory hog like Mobile Safari or Facebook I'll kill it. Otherwise I leave the apps and let the OS manage them.
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