a little self indulgent on the meta thing but since we've spoken about it and I adore it:
Logans relationship with Charles by the time of the Logan movie :)
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GOD OKAY BUCKLE UP. Because this is one of the only live action movies that I think really, really understood Logan and Charles alike. In the other movies, you tend to get one or the other right, but the Logan movie, and Deadpool and Wolverine, gets the both of them. I also take inspiration of their relationship from Evolution (my favourite piece of X-Men media of all time), and bits and pieces from WatXM and some comics. But the movie.
Strictly canon, Logan sees Charles as his father figure. Yes, he is at least a century older than Charles, but think about it. When Logan comes to the X-Men, he looks like he's about thirty, give or take, doesn't know exactly how old he is, and has MAYBE between five to fifteen years lived experiences that he remembers (the exact number differs between stories/runs/adaptations/etc. I usually go closer to fifteen to account for his rehabilitation with the Hudsons, then his time with Alpha Flight, then his time on the run). He has nothing. If you ask him, he is nothing.
But the X-Men - but Charles - don't see him that way. They give him a home in the literal sense, which was incredibly helpful. He had a safe place to sleep. He could shower regularly without worrying about water hook-ups. He had healthy food and had it regularly. He could wash his clothes. He had regular contact with trustworthy people - and other mutants, no less. He didn't have to run anymore.
That alone is huge. The fact that he had real, unconditional support for the first time in his memory - I just can't overstate how much that changed things for him. He'd never really had time to decompress and process and heal from the things that had been done to him. He'd been in survival mode. Even in Alpha Flight, he couldn't really relax. He tried, sure. He thought he'd settled in. Then things got bad, and they stayed bad, and only got worse. He owed them. He belonged to them. If he disagreed, he was ungrateful. If he left, he should be brought back and have the option to leave taken away. It's not good, y'all. But with the X-Men - at least early on (and in Evolution, again, please watch it if you haven't), if he wants to leave? Sure, Logan, have fun. We'll keep your room safe for you. See you when you get back. You want to argue with authority? Well, we'll argue back, but we won't punish you for it. You want a say in what happens and how we handle missions? Of course, we value your strategic insight. He's not just a weapon, or an animal, or a charity case, or a tragedy, or a monster with them.
Alpha Flight may have helped him be a human again, but the X-Men help him to be a person - Charles helps him to be a person - and he can't put a price on that.
There's also the fact that Charles reminds Logan of his dad. Not Thomas - being reminded of Thomas is a bad thing - but he reminds him of John. This is especially true in earlier comics and Evolution, but I think we see that in the Logan movie, too - and in X1 and X2. X3 had some weird moments for him and we won't talk about it.
In X1 and X2, Logan doesn't remember anything about his past. He doesn't remember John - but I think there's some sort of subconscious draw towards Charles, because he and John have a very similar personality. They're both good men who can be stern, but are generally kind and good-humored, protective of those around them, loyal to their families, upper-class, well-mannered and well educated, and even both had absolute awful fathers of their own. (Again, subject to writers.) It would've helped Logan to warm up to him instinctively, because for those who remember, baby James idolized, and idealized, his dad. John was one of the only adults who didn't treat him horribly. Even if he couldn't always make time for James, he did his best to give his son a good life. Compare that to Elizabeth, who ignored him at best and abused him at worst; his grandfather, who was both physically abusive and blamed James' health problems on him being 'soft', and Thomas, who- I mean, again, where the fuck do you start with Thomas? John was really the only adult - outside of the staff, and even then it was mostly Mrs Hopkins - who made James feel loved, or even just wanted. Combine that with the sudden, traumatic nature of his death, and it's safe to say Logan has daddy issues. Taking care of Charles - even if that just means being the guard/attack dog of the X-Men, or helping with the school - feels almost like a second chance. He can be the good enough son this time. He can keep his dad safe. He can be worthy of affection.
Now, this is where we get into another popular fan comment: Logan is like, a century and a half older than Charles, at least. How can he be his father figure?
Logan has no idea how old he is.
That's something I feel like canon doesn't fully appreciate a lot of the time - he has absolutely no memories at all. He doesn't know how old he is. He doesn't have any life experience beyond waking up in the forest with the Hudsons, which the movies says is roughly fifteen years - I go with about that, too. He looks roughly 30, give or take. That's all he has to go off of. When he meets Charles, as far as he knows, Charles is significantly older than him. By the time they figure out that's not the case, the dynamic is already established.
It's also interesting because Charles fills a lot of roles for Logan. Not only is he Logan's father figure, he's also his boss, his general, and his therapist. And, y'know, his friend. It's a lot. The lines get blurred, especially given the fact they also live together - different rooms, mind, but same building. They have most meals together. See each other on a daily occurrence. They also give each OTHER advice, since Logan's combative and tracking experience is second to basically none. Even the nickname that was originally meant to annoy him - Chuck - becomes a symbol of their friendship, with Charles bluntly informing other characters who try to call him that that only Logan gets to use that name for him.
All of this contributes to the frankly unhealthy level of loyalty Logan feels not just towards the team, but Charles, too. He'd do anything for Chuck. We see the culmination of that in the Logan movie.
For those who haven't seen it, spoilers ahead. But also watch Logan. It's a damn good movie.
So, if you listen to the radio when Charles, Logan, and Laura are driving, you hear that Charles' powers malfunctioning killed at least some of, if not all, of the X-Men. If you listen to the early discussions, and Charles talking to 24 when he thinks it's Logan, we know Charles doesn't remember that until right before he gets killed. If you really listen to the first conversation between Logan and Charles, you learn that Charles thinks Logan did something to get rid of the team.
Logan is letting him think that. Logan is taking the blame for losing the team, their home, and everything else, taking all of the pain and the guilt, so Charles doesn't have to. He would rather Charles hate and blame him than blame himself. That also says something about Logan's guilt complex, but that's a meta for another post.
There's also just the living situation. They're living in an abandoned iron foundry. Why? It's not because Logan thinks it's comfortable. It's not because he likes it there. It's because it's out of the way, and, as Donovan points out, the metal sphere Logan keeps Charles in can help mitigate the effects of his seizures. It doesn't contain it fully, but it helps. It's also supposed to help keep out any interference or outside thoughts Charles might get leaking into his perceptions - but that, too, only kind of works.
But it's better than nothing.
And when you compare Charles' little area to the rest of the foundry, it's actually kind of nice. He has plants. He has a little grow light to do gardening. He has space to move around in his chair. It's not NICE, but it's not terrible, either - it at least looks like they're trying. Compare that to Logan's own little room that barely fits his bed, has Mariko's sword on the wall, and that's about it. Logan works a shit job as a taxi driver that he usually seems pretty miserable to care for and provide for Charles (and Caliban), and essentially begs for drugs and bribes nurses at a hospital to get Charles what he needs. Compare that to the first aid Logan gives himself - which he needs without his healing factor. According to Caliban, Logan treats himself by wiping the pus from his infected hands onto his shirt and then bandaging them with... It might be gauze. Nothing that happens to him matters as long as Charles is taken care of, and this shows itself even more in Logan's plans for the Sun Seeker. As we see in the movie, Logan is saving up what little spare cash he has to buy a boat, and as Caliban says, this is for Logan and Charles. We also know that Logan was carrying an Adamantium bullet to kill himself. (This shouldn't work, but the XMCU doesn't understand Adamantium, or Logan's healing factor, so we ignore the discrepancies for story in this case. But it wouldn't work for multiple reasons.) As far as we know, judging by the conversations, his plan is to save up to buy a boat to take Charles away from society, to where he can safely be outside without them having to worry about his seizures hurting anyone, take care of him until he dies, and then Logan is going to kill himself in the middle of the ocean. He is alive only because Charles needs him. When Charles dies, Logan will have no reason to keep going (until Laura, of course). Logan doesn't want to be alive, every moment is physical agony, he can feel the Adamantium poisoning him from within, he's old, his senses don't work like they should, he lives in a dump, he hates his job, they barely have enough for food, he misses his team (his pack, his family), he doesn't really have any friends or support system (Caliban tries but they're both a mess), but he keeps going because Charles needs him, even if their relationship is horribly strained not just because of Charles' condition, but because Logan is letting Charles believe it's Logan's fault the team is gone so he doesn't get hurt by the truth.
You'll also notice that, while Logan is his usual snippy, argumentative self, he has a very hard time saying no to Charles. Charles wants to buy new clothes, fuck it, they buy new clothes. Charles wants to stop and help the family, fine, they stop. Charles wants to join them for dinner, fine, he wants to stay the night, fine. Logan caves on all of these even though he knows, and makes it clear that he knows, that they're all bad ideas and it's dangerous, because this is what Charles wants. He wants Charles' approval and he wants Charles to still want him around, because even centuries later, if he's not good enough, Logan thinks he'll be abandoned. So they stay at the farm house. And then Charles dies, and it ruins Logan - Laura becomes his hope, but he carries that guilt and grief until his death... Which, incidentally, is not far off.
I also want to say though, that it's not the Logan movie, but we do see how Logan WOULD have carried on without Charles in Deadpool and Wolverine - and the answer is, after a complete fucking meltdown from losing his family, Logan would try to live up to who they are. When he confronts Cassandra, it's not just the idea of being an X-Man that makes him choose the 'moral' path. He does it for Charles. Because Charles showed him even someone like him deserved love and a second chance. Charles wouldn't have let her die, and Logan wanted to be the man Charles believed he could be, even just once - and that one act of mercy, that one time he tried to honour Charles' legacy, gave him the chance to save the entire multiverse.
He kinda cares about Chuck or something, but don't tell nobody.