Logan is going to have a very hard time dealing with Soldier Boy, let's see:
In LS, Soldier Boy has the advantage, however, he doesn't use it as much as, for example, Homelander does, so it's not a big problem.
In AP/Durability, Soldier Boy practically destroys all of Logan's organs and tissues with each blow. Even if he regenerates and resists with his Stamina, Logan is going to be treated like a rag doll. Even with his bones and claws, he's going to have some difficulty withstanding the blows. And if Soldier Boy unleashes his blast on him, I think it would leave him incapacitated, even if it doesn't kill him.
So, in AP, Durability, and LS, Soldier Boy has a solid advantage to win... at the beginning. Logan can survive any beating Soldier Boy gives him thanks to his regeneration, and if he manages to prolong the fight, he'll surely be able to even the odds with his Attack Damage and perhaps even surpass him, although the AP gap is very high, but it's possible. Furthermore, Wolverine's claws should be able to damage Soldier Boy's defense without problems, and considering how skilled and experienced Logan is, it won't take him too long to counter it somehow.