This is normal behavior of the mouse.
The green battery light of your MK320 mouse will come on for a few seconds to indicate the battery condition and then go out. This will occur when you power on your mouse and when you replace the battery. The battery light goes out after a few seconds to conserve battery power.
If the above solutions do not resolve your problem, consult Knowledge Base article 16097 for instructions on contacting Logitech Technical Support.