Like Matthew, Luke records Jesus’ quotation from Deuteronomy 8:3 in response to the devil’s temptation. But the quotations differ:
Matthew 4:4: “Man shall not live on bread alone but on every word that proceeds out of the mouth of God.”
Luke 4:4: “Man shall not live on bread alone.”
What’s happened to the rest of the quotation? Luke 4:22 supplies the answer. When Jesus speaks in the synagogue at Nazareth, everyone is initially astonished at “the gracious words that were proceeding out of His mouth.”
(In Greek, the phrase “proceeding out of His mouth” in Luke 4:22 is slightly different from the quotation in Matthew 4:4; Matthew uses rhema instead of logos for “words” and dia rather than ek in the prepositional phrase “out of His mouth. The verb “proceed” and the word for “mouth” are the same in both passages, however, and the substance of the phrase is the same.)
Luke does not have Jesus say that life comes through Yahweh’s words. Instead, he tells us that the people marvel at the words of grace and life that come from Jesus’ own mouth. As He has so often in the early chapters of his gospel, Luke identifies Jesus as Yahweh.