What number on the scale should I log?

Weighing daily is fine and in fact I recommend people do it, BUT (and this is a big BUT), you do NOT pay attention to the day to day changes. Weigh daily right after you wake up after you pee, record those weights in a log. After 7 days, take an average. After a month compare the averages. This will show your actual progress. The problem with weekly weigh ins is if you happen to weigh in on a day where water retention is usually high or low due to sodium or glycogen or any of a million things, you don't have to freak about the massive shift in weight. The averages will account for the high days and the lows days and give you an idea of what you really weigh. After a month goes by, your averages should go in the downward direction.

i weigh daily, and only record every time i hit a new lowest number.

I weigh daily but don't change the log unless the weight has stayed the same for more than 3 days. My scale is cheap and the weight will sometimes go up and down within that time. So I wait until I see consistency.

I record the first morning weight every day. I don't flip out if it goes up or get too excited when it goes down.

But the first 30-40 pounds, I never weighed myself at all.

Some people like a weekly number. Everyone is different.

Do what you want. It's your weight loss journey!