John Bunyan in his book Pilgrim’s Progress has a main character, Hopeful, say it this way:
“He [Faithful, another character in the story] bid me go to him and see. Then I said it was presumption. He said, no, for I was invited to come. Then he gave me a book of Jesus’ inditing, to encourage me the more freely to come; and he said concerning that book, that every jot and tittle thereof stood firmer than heaven and earth. Then I asked him further how I must make my supplication to him; and he said, Go and thou shalt find him upon a mercy-seat, where he sits all year long to give pardon and forgiveness to them that come. I told him that I knew not what to say when I came; and he bid me say to this effect: God be merciful to me a sinner, and make me to know and believe in Jesus Christ; for I see, that if his righteousness had not been, or I have not faith in that righteousness, I am utterly cast away. Lord, I have heard that thou art a merciful God, and hast ordained that thy Son Jesus Christ should be the Savior of the world; and moreover, that thou art willing to bestow him upon a poor sinner as I am — and I am a sinner indeed. Lord, take therefore this opportunity and magnify thy grace in the salvation of my soul through thy Son Jesus Christ.”
Bunyan says that Hopeful did not understand at once, but soon he did and said:
“From all which, I gathered that I must look for righteousness in his person, and for satisfaction for my sins by his blood; that what he did in obedience to his Father’s law, and in submitting to the penalty thereof, was not for himself but for him that will accept it for his salvation, and be thankful.”
This is what “believing on the Lord Jesus” means. If a man has believed in this way, he has God’s promise that he is a Christian.
Thoughts developed and/or taken from the works of Dr. Francis A. Schaeffer, Trilogy – The God Who Is There