I have had this phone a week today. I had the Note 2 which the only thing bad I can say about it was the size. After checking into the S4 I really liked the phone. Well....it doesn't keep a charge but a couple hours at most even if not being used. As soon as I take it off the charger it starts going down. The phone gets so hot in just a few minutes that I can't even hold it to my face. I am about to return it and get something else but not really happy about that. I love the display and phone size but it doesn't help to have to charge it a few times a day.
If I was you, I'd take it back and ask for a new one. I have had mine since the day it was released and no issues. My charge also holds the whole day. I don't do much on my phone though, just facebook and a few websites, but tons of texting and picture messaging.
You need to check what programs are running. Bluetooth and GPS use battery. If you use wireless and have a weak signal, that uses battery too. There is an app called Battery Guru that monitors battery usage. There are many app killing/battery saving apps available. You need to be careful which you chose. With my previous phone, I downloaded an app killer that totally screwed up the phone.
It could just be your battery. My S4 lasts all day without modest use and little data usage.
Something is wrong with your phone. My S4 battery charge always lasts more than a day, even with decent usage, leaving GPS on, no special battery saver app, etc. I used to take all those precautions with previous phones but now it doesn't matter; S4 battery life should be very good for an android.
Well I am starting to see a trend here. Mine holds a charge, but it does get warm with regular use
and once so hot I literally thought it would self combust.