Acts 19:17

And this became known to all, both Jews and Greeks, that dwelt at Ephesus; and fear fell upon them all, and the name of Jesus was magnified.

The tremendous results which followed the discomfiture of the ambitious "seven sons" was brought about by the widespread interest in a place like Ephesus in magical arts of witchcraft, etc. In Shakespeare's COMEDY OF ERRORS (Acts 1, Scene 2, lines 97ff), one finds the following description of Ephesus:

They say this town is full of cozenage, As, nimble jugglers that deceive the eye, Dark-working sorcerers that change the mind, Soul-killing witches that deform the body, Disguised cheaters, prating mountebanks, And many such-like liberties of sin.

Thus it came about that in such a climate dominated by the works of darkness, the victory of the name of Jesus over the would-be exorcists was just the kind of thing to have produced the results Luke mentioned.