See his autobiography (1976).
Logan was born in Texarkana, Texas. His father died when Logan was only three, and his mother remarried six years later. He was reared in Shreveport, the seat of Caddo Parish and the largest city in north Louisiana. He attended Culver Military Academy in Culver, Indiana, where his stepfather served on the staff. 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 study in Moscow with Constantin Stanislavsky, and Logan left school without a diploma.
In 1942 Logan was drafted by the US Army. During his service in World War II, he acted as a public-relations and intelligence officer. When the war concluded he was discharged as a captain, and returned to Broadway. He married his second wife, actress Nedda Harrigan, in 1945; Logan's previous marriage, to actress Barbara O'Neil, a colleague of his at the University Players in the 1930s, had ended in divorce.
After the war, Logan directed the Broadway productions Annie Get Your Gun, John Loves Mary, Mister Roberts, South Pacific, and Fanny. He shared the 1950 Pulitzer Prize for Drama with Richard Rodgers and Oscar Hammerstein II for co-writing South Pacific. The show also earned him a Tony Award for Best Director. Despite his contributions to the musical, in their review the New York Times originally omitted his name as co-author, and the Pulitzer Prize committee initially awarded the prize to only Rodgers and Hammerstein. Although the mistakes were corrected, in his autobiography Logan wrote "I knew then why people fight so hard to have their names in proper type. It's not just ego or 'the principle of the thing,' it's possibly another job or a better salary. It's reassurance. My name had been so minimized that I lived through years of having people praise 'South Pacific' in my presence without knowing I had had anything to do with."
Logan cowrote, coproduced, and directed the 1952 musical Wish You Were Here. After the show was not initially successful, Logan quickly wrote 54 new pages of material, and by the ninth performance the show looked new. In its fourth week of release, the show sold out, and continued to offer sell-out performance for the next two years.
His later Broadway musicals All-American (1962) and Mr. President (1962) and the films of Lerner and Loewe's Camelot (1967), and Paint Your Wagon (1969) were less acclaimed. Logan's 1976 autobiography Josh: My Up-and-Down, In-and-Out Life talks frankly about his bipolar disorder. He appeared with his wife in the 1977 nightclub revue Musical Moments, featuring Logan's most popular Broadway numbers. He published Movie Stars, Real People, and Me in 1978. From 1983-1986, he taught theater at Florida Atlantic University in Boca Raton. He was also responsible for bringing Carol Channing to Broadway in Lend an Ear!.