alright wrote
so if I ran 65 mins, rather than worrying about "oh my god, was it 10 miles, or 10.3?" just put down 10 miles
If it is one day it doesn't matter much. But if you do this every day, the cumulative effect on your body is different. What if it was 10 vs 11 miles or 11.5? That's a significant difference that adds up over time.
Do you understand the purpose of a training log? It is a learning tool. You write down what you did, how you felt, how hard you ran, etc etc. Then, after some time passes, you go back over the log and look for patterns. You see what worked and what didn't work. This informs your future training and allows you to improve it by ceasing to do things that caused problems.
It is true that whether you keep a log or not you have done the miles. The only difference is that the guy who keeps the log can learn from the past. Because he has an accurate record, not fuzzy memories of "about" what he did. You're right that you don't really need it. But if you are going to keep a log, keep it accurately. Don't make shit up. That's worse than not keeping it at all.
If you cannot know the distances you run that's a different story. But if you can, it is senseless to hide information from yourself.