Born Myer Alexander on July 2, 1910 in Seattle, Washington, Alexander was a conductor, arranger, and composer of scores for TV, radio, and film. Known as "Jeff Alexander" he should be best known for having arranged Sinatra's historically important recording of The House I Live In (1945); however, credit has always been given to Axel Stordahl. During the research of the music for publication, Jazz Lines Publications' discovered that the arrangement was actually composed by Alexander.

Alexander began performing in his teens as a singer and shortly thereafter began playing piano and composing music. Relocating to New York City in 1939, he began composing music for radio programs, including Camel Caravan, The Lucky Strike Show, and Amos 'n' Andy. He also led chorus groups for radio performances and studio recordings. In 1947 he moved to Los Angeles to embark upon a film scoring career. He would go on to score many of Elvis Presley's films, including Jailhouse Rock (1957), Kid Galahad (1962), Double Trouble (1967), Clambake (1967), and Speedway (1968). He also composed the scores to over 30 films, including The Tender Trap (1955), Ransom! (1956), The Wings of Eagles (1957), The Sheepman (1958), Party Girl (1958), Ask Any Girl (1959), The Mating Game (1959), The Gazebo (1959), All the Fine Young Cannibals (1960), The George Raft Story (1961), The Rounders (1965), Day of the Evil Gun (1968), Support Your Local Sheriff! (1969) and Dirty Dingus Magee (1970).

In 1956, Alexander contributed the tone poems Yellow and Brown to the album Frank Sinatra Conducts Tone Poems of Color. He also composed a symphony and other classical pieces. In 1989 Alexander died of cancer.