It's not necessary to keep a log to focus on increasing the difficulty of workout. Over a period of time your body gets used to a certain weight and at that time you'll feel that it's too easy to lift that certain weight. You then increase the weight.

I train with weights, and I they certainly do not weigh the same on different days. Some days are better than others, a log helps keep me on track.

Of course that's me. It might work for you though. In my coaching experience I find, that most people are not objective enough (or have a good enough memory) to work like that.

Tracking calories on paper vs in your head is a similar objective/subjective mess for most people. The mind is very good at deceiving itself.

I train with weights as well, but I do a whole body routine regularly. It's not very hard to remember the weights (at least the ballpark). I think if you do isolated exercises it'll be harder to remember the weights and a log will certainly help.

Well, a log is the easiest way to remind yourself what you weight you lifted a few days ago, if you are having trouble remembering. But no, it's not required.