Do you need a log book for a commercial vehicle under 26001 lbs?

Depends. If its transporting a quantity of hazardous materials which requires the display of placards or else is a bus designed to transport more than 15 persons (including the driver), then it's a commercial vehicle at any weight which requires the appropriate class of CDL and endorsements (for a vehicle under 26,001 GVWR/GCWR, it's a minimum Class C CDL), and all logging and HOS regulations apply.