Luke joined the Apostle Paul sometime around the events of Acts 16:10
When you read the book of Acts you discover that Luke, the author, was one of the Apostle Paul’s traveling companions. But if you read closely you will find that Luke didn’t join the Apostle Paul on his journeys until around Acts 16:10.
We determine that from Luke’s usage of pronouns. Up until around Acts 16, Luke uses the pronoun, they, when referring to Paul and his companions. This indicates that he is reporting on something he was not a part of, or not present.
And as they went through the cities, they delivered to them the decrees to keep…. —Acts 16:4
In Acts 16:10 we observe a change in the use of pronouns to we. This indicates that Luke was now a part of Paul’s traveling group.
And after he had seen the vision, immediately we endeavored to go into Macedonia, concluding that the Lord had called us to preach the Gospel to them. —Acts 16:10
There seems to be some consensus among scholars that this is the approximate time Luke, the author of the book of Acts, joined the Apostle Paul on his journeys.