Note 3 battery life took a nose dive

Kitkit has a known bug that destroys battery life on Note 3's and only way to avoid it is to not upgrade to 4.4
 
Reinforcing what others here have shared. GSAM's App Sucker feature is a good way to determine exactly which system process or app may be contributing to the problem.
 
You can also download easy battery saver or juice defender. They shut data off when you're not using it and periodically turn it on to sync data from your apps. For the most part in doesn't affect regular use of the phone unless you absolutely have to receive your email as soon as it comes in for instance.

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