Some years ago while attending BYU, I took a look at the Harmony of the Gospels to see what each of them included that was unique. That's when it began to dawn on me -- Luke was a Jew, writing to the learned Jews. His purpose was to show the learned Jews that Christ fulfilled every jot and tittle of the Mosaic Law and His grand mission was to bring salvation to all of mankind, yes, the Gentiles. Matthew, on the other hand, is writing to convince the Jews that Christ was the fulfillment of the prophecies, and he's very good at including everything that is a fulfillment of prophecy.
Now let's take a good look at Luke, and his dual purpose -- the Law and the Universal Gospel. Remember the Pharisees and Scribes were OCD about the law. And the Pharisees and the Scribes did not want the Gentiles to be part of the Gospel -- probably none of the Jews did, but the Phariesees and Scribes even more so. They didn't even want Jesus associating with sinners. They had nothing to do with the Samaritans.
First let's look at Luke's opening paragraph, which I think should make it pretty clear that Luke had first-hand knowledge of things "from the beginning."
1 Forasmuch as many have taken in hand to set forth in order a declaration of those things which are most surely believed among us,
2 Even as they delivered them unto us, which from the beginning were eyewitnesses, and ministers of the word;
3 It seemed good to me also, having had perfect understanding of all things from the very first, to write unto thee in order, most excellent Theophilus,
4 That thou mightiest know the certainty of those things, wherein thou hast been instructed.
What can make it more clear than that -- Luke was an eye-witness. Look at Luke 2 -- every point of the Mosaic Law is included. And Luke says, the Law -- he doesn't leave it to chance that the Pharisees and Scribes are going to miss it. It wasn't important for the Gentiles to know that the Mosaic Law was perfectly followed because they were never asked to follow the Mosaic Law.
And all of the witnesses of Christ's birth are Jews -- the Shepherds, Simeon, and Anna.
Luke also on occasion calls attention to prophecies. Unfortunately one of his direct quotes from Isaiah is mostly missing from the KJV, but restored by Joseph Smith. This is what is in the JST, and the italics is what was restored. Chapter 3
4 As it is written in the book of the prophet Esaias; and these are the words, saying, The voice of one crying in the wilderness, Prepare ye the way of the Lord, and make his paths straight.
5 For behold, and lo, he shall come, as it is written in the book of the prophets, to take away the sins of the world, and to bring salvation unto the heathen nations, to gather together those who are lost, who are of the sheepfold of Israel;
6 Yea, even the dispersed and afflicted; and also to prepare the way, and make possible the preaching of the gospel unto the Gentiles;
7 And to be a light unto all who sit in darkness, unto the uttermost parts of the earth; to bring to pass the resurrection from the dead, and to ascend up on high, to dwell on the right hand of the Father,
8 Until the fullness of time, and the law and the testimony shall be sealed, and the keys of the kingdom shall be delivered up again unto the Father;
9 To administer justice unto all; to come down in judgment upon all, and to convince all the ungodly of their ungodly deeds, which they have committed; and all this in the day that he shall come;
10 For it is a day of power; yea, every valley shall be filled, and every mountain and hill shall be brought low; the crooked shall be made straight, and the rough ways made smooth;
11 And all flesh shall see the salvation of God.
This, to me, is a well-learned Jew telling other well-learned Jews that the Gospel is not only for the Jews, but also for the Gentiles -- it is Universal. And this is Luke-s dual purpose -- to show the exact fulfillment of the Law and to prepare the Jews for the Gospel going to the Gentiles.
While Matthew mentions Lost Sheep, Luke's parable of the Lost Sheep is much more involved, followed by the Parable of the Lost Coin, the Parable of the Prodigal Son -- all of which involve recovery of those that have been lost, and that would be the Jews. He also told the Parable of the Unjust Steward, and the Parable of Lazarus and the Rich Man, which no other Gospel Writers include and which I believe were directed particularly to the Pharisees.
Luke is the only Gospel Writer that includes the rejection of Christ by the people of his hometown, Nazareth.
One more thing, which can only be found in the JST of Luke. This is what it says in Luke 3:12-13:
12 Then came also publicans to be baptized, and said unto him, Master, what shall we do?
13 And he said unto them, Exact no more than that which is appointed you.
This is what Joseph Smith added to it:
19 For it is well known unto you, Theophilus, that after the manner of the Jews, and according to the custom of their law in receiving money into the treasury, that out of the abundance which was received, was appointed unto the poor, every man his portion;
20 And after this manner did the publicans also, wherefore John said unto them, Exact no more than that which is appointed you.
Luke knew well the "customs of their law" because he was one of them.
In summary, I believe Luke had a two-fold purpose in writing his Gospel - to convince the Jews that Christ was the fulfillment of the Law and the Gospel was going to be taken also to the Gentiles. Luke would later document the Gospel being first taken to the Jews and then to Gentiles in his Book of Acts.
