Logan sanders is a representation of Thomas logic yes, but that does not mean that he is only logic. All of thomas' sides represent more than one thing, and even if Logan was only logic, logic and emotion are so tightly linked that logic could not exist without emotions (outside emotions included).
Logan's entire job is trying to make sense of emotions, logic is knowing facts, knowing why something does what it does and using that to figure something out. Apply this to humans and the root of the action is almost always based off an emotion they felt. Now, if we apply that to Logan, he most likely wouldn't do half the things he's done if he didn't have emotions, here's a list off the top of my head.
-"FALSEHOOD" not a thing
-he would have left in Moving On (Part 1)
-he would not be excited about Thomas considering learning about astronomy
-"WE👏GET👏IT👏YOURE👏A👏DOR👏ABLE" would never happen
-he would not feel bad or try to make up for when he messed up in "The MIND vs The HEART", he would acknowledge the mistake and learn from it, then move on with no second thoughts.
There are even more examples, those are just the ones I thought of first.
There are a lot more reason which I'll probably add later, but it's 1 am right now and I'm tired.
