You start Uber for the day, write down kms/date/time etc in your log book. Do your uber trips etc as per normal, then when you finish Uber write down your ending kms/date/time. You don't need to keep a logbook for EACH uber trip, but you do need to keep one for each uber 'shift' as such.
So if you head out to do uber once a day for 8 hours straight and do 30 uber trips in that time, you only need one logbook entry, not 30.
From here you can work out that you did say 10,000kms over that 12 weeks, of which 8,000kms had been for uber aka business. So your business usage rate is 80% for tax purposes.