In choosing to 'pit' yourself against Luke, you also pit yourself against Matthew, whose Gospel supplies necessary and complimentary information on the nativity and genealogy of Jesus.
Both Matthew and Luke mention that Herod the Great was the king at the time of the birth of Jesus. Matthew goes on to tell us about the death of Herod, and the return of Joseph and Mary from Egypt, their place of refuge after the massacre of the children at Bethlehem.
Matthew 2:19-23.
'But when Herod was dead, behold, an angel of the Lord appeareth in a dream to Joseph in Egypt,
Saying, Arise, and take the young child and his mother, and go into the land of Israel: for they are dead which sought the young child's life.
And he arose, and took the young child and his mother, and came into the land of Israel.
But when he heard that Archelaus did reign in Judea in the room of his father Herod, he was afraid to go thither: notwithstanding, being warned of God in a dream, he turned aside into the parts of Galilee:
And he came and dwelt in a city called Nazareth: that it might be fulfilled which was spoken by the prophets, He shall be called a Nazarene.'
Matthew and Luke together provide us with key events, which, when taken together, can be formed into a chronology.
1. Luke mentions Caesar Augustus [Luke 2:1] From other sources we know that 'Octavian' was Emperor between 27 BCE and 14 CE.
2. Herod the Great was king in Judea ( poss. 37 BCE - 4 BCE) during the reign of Octavian. Judea became a Roman province with a 'client king' after the capture of Jerusalem by Gnaeus Pompeius Magnus (Pompey) in 63 BCE. Following the death of Herod, his son Archelaus reigned. Direct Roman rule began in 6 CE when Archelaus was deposed as king of Judea (by Augustus) and sent to Gaul.
3. It was during the time that Octavian was Caesar that Cyrenius (Quirinius), the governor of Syria, taxed his territory. Luke adds, '(And this was the first made when Cyrenius was governor of Syria)' Note that Luke would not have found it necessary to add these words if there was ONLY ONE CENSUS during Cyrenius' governorship of Syria!
4. Josephus records that there was a census in Judea in 6CE, which must have been the second under Cyrenius. The big difference between the two censuses under Cyrenius is that the first was conducted under Jewish methods, by Herod the Great, the second under Roman method. This is why citizens under Herod, who followed Jewish custom, were told to return to their ancestral homes, which for the progeny of Jesse/David was Bethlehem.
5. In chapter 3 of Luke, we are given additional information that takes us from Jesus' birth to his manhood. 'Now in the fifteenth year of the reign of Tiberius Caesar, Pontius Pilate being governor of Judea, and Herod being tetrarch of Galilee, and his brother Philip tetrach of Iturea and of the region of Trachonitis, and Lysanias the tetrarch of Abilene, Annas and Caiaiphas being the high priests...'
Tiberius became emperor in 14 CE and remained emperor until 37 CE. So Luke is talking about 29 CE, maybe 30 CE, when he mentions 'the fifteenth year of the reign of Tiberius Caesar'. At this time Pontius Pilate is governor of Judea (governor between 26-36 CE).
6. In Luke 3:23 we are told that Jesus was 'about thirty years of age' when he was baptised by John. Since Jesus was baptised in the fifteenth year of Tiberius' reign, we can calculate that Jesus was born about the year 5/4 BCE. This is a full ten years before the second census of Cyrenius (6 CE).