The GREEN output is for STEREO output to your FRONT speakers.
Your logitech speakers will have a MANUAL explaining the setup.
Assuming you ONLY have the red/white input then it's likely you have STEREO speakers. In which case you would use this wire:
PC sound card end-> GREEN, 3.5mm jack
SPEAKER end-> two jacks, one red and one white
*You'll have a main speaker (or subwoofer) with the amplifier; this is connected to the sound card. It will connect with a wire or wires to the other speaker or speakers.