Yes you should have your headlights on during the fog.
Use your low beam lights in the fog which should be good enough and if your car or vehicle has fog lights you should use the fog lights to help you see better in the fog.
I added fog lights to my car and truck and it really helps when driving in the fog.
If you're driving in fog at night be sure to really slow down at least 10 to 15 mph below the speed limit and then you may need to use your high beam headlights until an oncoming vehicle comes.
Then turn the high beams off and wait until the oncoming vehicle gets past you and then turn the high beams back on which is what I do.
Even during the day with fog you need to slow down some because fog reduces your visibility.