Answer: The purpose of St. Luke writing the St. Luke's Gospel can be is seen from Luke 1 : 1-4, which is that through his well researched and accurate account of the events of Christ's ministry; all he began to do and teach, a brother, Theophilus, can be convinced in the truth of what he has been taught in the doctrine of the faith in Christ Jesus.
Also, the purpose of writing the Book of Acts as seen in Acts 1 : 1-8 was to narrate to the same brother Theophilus the post-resurrection events which are: how the Apostles were baptised of the Holy Spirit; instructed and sent forth by the resurrected Christ to preach the gospel to all the world; the birth of the Church by reason of Christ's resurrection and the given Holy Spirit; and the acts of the Apostle in the ministry (which includes miracles, signs and wonders).
Explanation: Luke wanted to convince Theoplilus on the doctrine of faith he had received, therefore, he wrote both books (Luke's Gospel and the Book of Acts) as a narration of both the events of Christ's ministry on earth till his death and the acts of the apostles in obedience to the instruction of Christ after he ascended to heaven. He believed his account to be very accurate and well researched because they were according to the accounts of the events told by the very eye witnesses of Christ's ministry (primarily the Apostles or the Twelve), which validates its accuracy since these events were experienced first hand by the eye witnesses.
I hope this helps?