I've logged my workouts for 20 years. I have more than 10 years of journals stacked in my garage.
I've worked out for periods without them, but I find I make GAINS better when I use them. They are motivation. If I did 8 reps at a weight last week I'm going to push to get 9 cleans ones this week.
Workout logs let you see improvement every week.
When I train without them I THINK I keep track of everything in my head, but I don't really track these small gains the same way.
Also, logs have let me spot injuries coming. If I see a pattern of the weight/reps going down... I know something is wrong.