Logan began his Broadway career as an actor in Carry Nation in 1932. He was also in I Was Waiting for You (1933).

Back on Broadway he staged It's You I Want (1935) and To See Ourselves (1935) and was stage manager for Most of the Game (1935). He staged Hell Freezes Over (1935-36) and returned to acting with A Room in Red and White (1936).

He went to Hollywood where he did some dialogue directing on The Garden of Allah (1936), History Is Made at Night (1937), and Suez (1938). Logan was given the chance to co-direct a feature film, I Met My Love Again (1938) for Walter Wanger.

He directed Knickerbocker Holiday (1938), Stars In Your Eyes (1939), Osborn's Morning's at Seven (1939-40), Two For the Show (1940), and Higher and Higher (1940, 84 performances). None of these was a break out success but his revival of Charley's Aunt (1940-41) went for 233 performances, and the Hart-Rodgers musical By Jupiter (1942-43) with Ray Bolger went for 427 performances.

Logan was married briefly (1939–1940) to Actress Barbara O'Neil. After the divorce, he was married to Nedda Harrigan from 1945 until his death from progressive supranuclear palsy (PSP) in New York City in 1988.

In 1942, Logan was drafted by the U.S. Army. During his Service in World War II, he acted as a public-relations and intelligence officer. Logan was selected to become an assistant Director of Irving Berlin's This Is the Army and when in Europe organised "jeep shows" of entertainers serving as soldier doing their shows near the front lines.

When the war concluded he was discharged with the rank of 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.

Logan's post war directing career got off to a brilliant start with the musical Annie Get Your Gun (1946-49) which ran for 1147 performances.

He followed it with Anita Loos' Happy Birthday (1948, 563 performances), and Norman Krasna's John Loves Mary (1948-49, 423 performances). Logan's golden run continued with Mister Roberts (1948-51) which he co-wrote as well as directed; it ran for 1157 performances and earned him a Tony Award.

Then he directed and co-wrote South Pacific (1949-54) which went for 1925 performances. Logan 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 it."

Logan wrote, produced and directed The Wisteria Tree (1950), an adaptation of The Cherry Orchard which was a minor success.

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 performances for the next two years.

He had another success with Picnic (1953-54), which went for 477 performances. Krasna's Kind Sir (1953-54) lasted 166 performances, but Fanny (1953-54) which Logan cowrote, coproduced and directed, ran 888 performances.

Logan directed the film adaptation of his own Picnic (1955), which was popular and earned Logan an Oscar nomination. His next movie, Bus Stop (1956) with Marilyn Monroe, was another hit.

He went to Japan to direct Marlon Brando in Sayonara (1957), which earned hima second Oscar nomination for Best Director. Then he did the big screen version of South Pacific (1958).

Logan went back to Broadway and directed Blue Denim (1958, 166 performances) and the hugely popular The World of Suzie Wong (1958-60) (508 performances). He produced Epitaph for George Dillon (1958).

Logan returned to Hollywood with Tall Story (1960) which introduced Jane Fonda to cinema audiences. Back on Broadway he directed There Was a Little Girl (1960), his first theatre flop in a long time, running 16 performances. In Hollywood he did the big screen adaptation of Fanny (1961).

In 1961, he was a member of the jury at the 2nd Moscow International Film Festival.

Logan continued to alternate Broadway and Hollywood for the rest of the 60s. He did the Brodway musicals All American (1962, 86 performances) and Mr. President (1962-63, 265 performances), and Tiger, Tiger Burning Bright (1962-63, 33 performances), then made the film Ensign Pulver (1964).

After Ready When You Are, C.B.! (1964-65, 80 performances), he did the films of Lerner and Loewe's Camelot (1967), and Paint Your Wagon (1969). Back on Broadway he did Look to the Lilies (1970, 31 performances).

Logan experienced mood fluctuations for many years, which in the 1970s Psychiatrist Ronald R. Fieve treated with lithium, and the two appeared on TV talk shows extolling its virtues.

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.

In 1979 he produced Larry Cohen's Trick on Broadway. He directed Horowitz and Mrs. Washington (1980) which ran 6 performances.

From 1983–1986, he taught theater at Florida Atlantic University in Boca Raton, Florida. He was also responsible for bringing Carol Channing to Broadway in Lend an Ear!.

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.