If your phone shuts down with more than 10% charge remaining on the estimate, then you have a voltage / capacity issue in the battery and should have the device tested with Apple diagnostics to check the health of the battery. Think of a small town with a medium sized reservoir and a small water tower. The reservoir level can have water, but if it’s not pressurized (voltage) in the tower, the system can ask for more water (current) than the tower can supply. For a shower, the water just stops, but for an iPhone, it shuts off when the voltage drops too quickly.
The diagnostics read the low battery logs and compare them with thousands of other logs and Apple engineering standards to tell if your battery needs to be replaced.
Technically, those log files are total benefit with no downside. Your phone is out of juice and will shut down anyway. The OS is logging the data so you can take action on it if you choose. They don't cause anything other than logging what led up to the low power condition.
I watch my LowBatteryLog-YYYY-MM-DD-hhmmss.plist log files quite regularly to know when to seek service for my battery.
Technically, the lithium polymer cells can exhibit voltage drops when they are out of specification but not yet failing or in a state where they should be shut off and not used again.
Looking more deeply into your two posted logs, I would focus on these lines primarily:
Awake Time: 03:45:27 (13526)
Standby Time: 04:55:29 (17729)
Partial Charge: 0
Capacity: 0
Voltage: 3469 mV
When Partial Charge is 1 - that means that you plugged it in and the device received a charge between when it was full and first removed from power and it eventually shut itself off due to low power detection. In those cases, I really only focus on the Voltage - knowing when the device decided to preserve the remaining voltage for standby and battery protection.
The log above shows a very short awake time and indicates a likelihood that the battery isn't providing the correct duration and amount of power. Even if the CPU is full use, all radios are on, speaker cranking, brightness max - I expect 4 to 5 hours on most devices.
Unless that 3h45m run was a rare occurrence, my estimate is you need a hardware repair as sleeping more often will allow the sleep time to increase, but never increase the active time.
For newer iOS hardware (think iPhone 5 to 10) the voltage to available current to available capacity can also be due to aging, power management software or hardware issues.
Are you using IOS 7? When using IOS 7 be sure to check your apps which are running in the background, but normally these apps are also closed after some time in standby. Be sure to update your iPhone to IOS 7.1. When you say your iPhone/battery heats up when loading there is surely something wrong with your battery. Your USB transformer might heat up a little, but never had an iPhone heating up when charging. Contact Apple Support, or your reseller. Probably your battery needs replacement.
Best Answer
It's possible that the battery health has degraded over time. I would recommend you install Battery Life in your iPhone to check the battery health. If the battery health is below 80%, then you should consider replacing the battery. I replaced my iPhone 5's battery after 3-4 years after it hit 65% battery health. You should do the same. I was able to find a 3rd party battery and replace it myself. The process took me 30-40 mins. Also, go to the Battery section in the Settings app and see which apps are the biggest battery users.