There are a few ways to get the logcat. The user can run monitor.bat (as shown above; found in android-sdk-windows\tools)and select log messages to save, or the following commands with adb.exe may be used:
Clears the log:
adb logcat -c
Shows log messages to screen (Ctrl-C to stop):
adb logcat
Dump all log messages to a file:
adb logcat > log.txt
Dump just UE4 tagged messages to a file:
adb logcat > logUE4.txt
NOTE: for the last two command you will see nothing and need to press Ctrl-C to quit when done to complete writing to the file.