I know his name is Logan, but I don't actually know if it's his first name, or last name, or if he even has a first and a last name. To my understanding he ran away from his parents and chose to name himself Logan, but I'm fairly new to the Marvel universe and I could very well be 100% wrong.
He was born as James Howlett, but he adopted the nickname 'Logan' to keep his identity secret. (Logan was the last name of his groundskeeper). Rose was the one who gave him the name.
(Jemas, Bill, Quesada, Joe, Jenkins, Paul (w). Origin (2001–2002), Marvel Comics)
Born the second son of wealthy landowners John and Elizabeth Howlett in Alberta, Canada during the late 19th Century [...] James also assumed the name of "Logan" in order to hide his identity.