How do we know that the Luke 3 lineage is Mary's? We do not know it for certain, but that conclusion is the most reasonable. One factor is, again, the purpose of this particular gospel. Luke wrote primarily to Gentiles, and he stresses Jesus' humanity throughout his book. The evangelist thus gives our Savior's natural, biological family tree to show He shares humanness with the common man. He is not just the Jews' Messiah, but He is also the Gentiles' Messiah! So Luke's genealogy goes all the way back to Adam, rather than stopping at Abraham as Matthew's does.
Another factor is that Luke had to deal with a virgin birth. What a unique situation for a genealogist! Luke had to determine, therefore, what points would matter to a Gentile. Would he be concerned with Jesus' Davidic ancestry? Not initially. Would he care that Jesus is a Jew and an Israelite? Maybe. Would he desire to know if Jesus was a man like he was? Certainly! Thus, Luke would record a line of descent that showed His universality to every man, and this would go through Mary, Jesus' link to humanity.
Some raise objections to this on the basis of verse 23, particularly because it says, "Joseph, the son of Heli." Notice, though, that Luke does not use the word "begot" as Matthew does. In fact, he uses no word at all, just a marker to denote possession. So the phrase literally says, "Joseph, of Heli."
Some say, then, that this connotes a levirate marriage because Matthew says Joseph's father was Jacob. Levirate marriage, however, was fairly rare, so this is an unlikely stretch. Others argue that this is Jesus' "priestly" lineage, but this is even less probable, since it shows Judah, not Levi, as an ancestor (see Hebrews 7:14).
Bullinger, in his Companion Bible, gives a more likely explanation: "Joseph was begotten by Jacob, and was his natural son (Matt. 1:16). He could be the legal son of Heli, therefore, only by marriage with Heli's daughter (Mary), and be reckoned so according to law." At that time, Jewish law traced inheritance and descent through the male, not the female line. Thus, Luke 3:23 would be clearer if translated as, "Joseph, the son-in-law of Heli," or "Joseph, the legal son of Heli."
No matter which we choose, it traces Heli's line from that point on back to Nathan, the son of David. There is no stigma or disqualification in Solomon's name being absent from the list. In messianic terms, David's name is the vital one.
Richard T. Ritenbaugh