The teaching of the Day from the Book of Concord:

Thus in Luke 7:47 Christ says: Her sins, which are many, are forgiven, for she loved much. For Christ interprets Himself [this very passage] when He adds: Thy faith hath saved thee. Christ, therefore, did not mean that the woman, by that work of love, had merited the remission of sins. For that is the reason He says: Thy faith hath saved thee. 32] But faith is that which freely apprehends God's mercy on account of God's Word [which relies upon God's mercy and Word, and not upon one's own work]. If any one denies that this is faith [if any one imagines that he can rely at the same time upon God and his own works], he does not understand at all 33] what faith is. [For the terrified conscience is not satisfied with its own works, but must cry after mercy, and is comforted and encouraged alone by God's Word.] And the narrative itself shows in this passage what that is which He calls love. The woman came with the opinion concerning Christ that with Him the remission of sins should be sought. This worship is the highest worship of Christ. Nothing greater could she ascribe to Christ. To seek from Him the remission of sins was truly to acknowledge the Messiah. Now, thus to think of Christ, thus to worship Him, thus to embrace Him, is truly to believe. Christ, moreover, employed the word "love" not towards the woman, but against the Pharisee, because He contrasted the entire worship of the Pharisee with the entire worship of the woman. He reproved the Pharisee because he did not acknowledge that He was the Messiah, although he rendered Him the outward offices due to a guest and a great and holy man. He points to the woman and praises her worship, ointment, tears, etc., all of which were signs of faith and a confession, namely, that with Christ she sought the remission of sins. It is indeed a great example, which, not without reason, moved Christ to reprove the Pharisee, who was a wise and honorable man, but not a believer. He charges him with impiety, and admonishes him by the example of the woman, showing thereby that it is disgraceful to him, that, while an unlearned woman believes God, he, a doctor of the Law, does not believe, does not acknowledge the Messiah, and does not seek from Him remission of sins and salvation.