When Was The Gospel of Luke Written?

It appears that Paul quoted from the book of Luke which would date Luke prior to Paul's death. 64-67 Ad

1 Tim. 5:18 For the Scripture says, “You shall not muzzle the ox while he is threshing,” and “The laborer is worthy of his wages.” 19 Do not receive an accusation against an elder except on the basis of two or three witnesses.

And

Luke 10:5 Whatever house you enter, first say, ‘Peace be to this house.’ 6 If a man of peace is there, your peace will rest on him; but if not, it will return to you. 7 Stay in that house, eating and drinking what they give you; for the laborer is worthy of his wages.

What changes whether it was written 50,60, 80 or 120 ad?

Why the Gospel of Luke belongs in the Bible

Regardless of the dating, the Gospel of Luke belongs to the Bible because it contains authoritative teaching of Jesus to guide Christians as well as authoritative reports of who Jesus is (how he was incarnated into a baby, how his divinity was affirmed from heaven in baptism and transfiguration) and what he did (the 3 year public ministry of healing, death, resurrection, and ascension).

Why should we believe the story as true? It's because the author claimed to write it as history (history in the ancient 1st century sense, not too rigid as the modern historical method) shown by how the author wrote explicitly in his statement of purpose and his statement of method of inquiry (Lk 1:1-4, NLT):

Many people have set out to write accounts about the events that have been fulfilled among us. They used the eyewitness reports circulating among us from the early disciples. Having carefully investigated everything from the beginning, I also have decided to write an accurate account for you, most honorable Theophilus, so you can be certain of the truth of everything you were taught.

Why should we trust the author? One explanation is because the early church network had deliberated over 100-200 years over the provenance of the writing (i.e. tracing it to Luke the physician) and because the early church believed the apostles to provide trustworthy witness of Jesus, the revelation from God.

Traditional dating (early 60s) vs. liberal dating (80 to 2nd century)

The key reason for liberal dating is their presumption that it has to be written after some prophecies about the fall of Jerusalem (A.D. 70) came true. They also doubted that Luke, the physician accompanying St. Paul in his missionary journeys, was the author.

Below is a typical reasoning to date Luke to early 60s, proposed by respected evangelical scholars, from the highly acclaimed ESV Study Bible:

The earliest possible date of Luke-Acts is immediately after the events that Luke recorded in Acts 28, c. A.D. 62. In fact, Luke could have been written slightly earlier, and Acts could have been completed at that time. The specific date centers on two questions: would Luke have added to his Gospel later, and did he make use of the Gospel of Mark in writing his own Gospel? If Luke wrote Luke-Acts after the martyrdom of Paul (c. A.D. 64–67), some have suggested that the omission of the details of Paul’s trial and death seems strange (see Introduction to Acts: Date). In addition, Luke makes no mention of the terrible persecution under Nero in A.D. 65 but gives a very positive picture of Paul preaching the gospel in Rome for two years “with all boldness and without hindrance” (Acts 28:31), so he must have written sometime before 65. (Those who hold to a later date for Luke reply that ending with Paul preaching in Rome is natural, because it shows that the preaching “to the end of the earth” in Acts 1:8 has been fulfilled. Since a primary purpose of Luke-Acts is to tell the story of how the gospel spread to the Gentile capital of Rome, when that purpose is accomplished the story could come to a natural end, regardless of what happens to Paul.)

Concerning the relationship of Luke to Mark, the great majority of scholars believe that Luke made use of Mark in writing his Gospel. There is no real difficulty in dating Mark in the mid- to late-50s A.D. (see Introduction to Mark: Date and Location), which would allow for a date of Luke in the early 60s. Other scholars dispute this and claim that the predictions of the fall of Jerusalem in Luke 21:20–24 are so vivid that they could have been written only after the fact, which means that Luke and Acts were written after A.D. 70. These scholars would then date Mark somewhat before Luke, in the late 60s (after Peter’s death). However, many evangelical scholars, who consider Luke 21:20–24 a predictive prophecy by Jesus, would hold to a date for Luke in the early 60s.