Logan was born in Texarkana, Texas, the son of Susan (née Nabors) and Joshua Lockwood Logan. When he was three years old his father committed suicide. Logan, his mother, and younger sister, Mary Lee, then moved to his maternal grandparents’ home in Mansfield, Louisiana, which Logan used forty years later as the setting for his play The Wisteria Trees. Logan's mother remarried six years after his father's death and he then attended Culver Military Academy in Culver, Indiana, where his stepfather served on the staff as a Teacher. At school, he experienced his first drama class and felt at home. After his high school graduation he attended Princeton University. At Princeton, he was involved with the intercollegiate summer stock company, known as the University Players, with fellow student James Stewart and also non-student Henry Fonda. During his senior year he served as President of the Princeton Triangle Club. Before his graduation he won a scholarship to travel to Moscow to observe the rehearsals of Konstantin Stanislavski, and Logan left school without a diploma.