The impact of prayer in Jesus' name
Jesus on the efficacy of prayer in faith
When they came to the disciples, they saw a large crowd around them and experts in the law arguing with them. When the whole crowd saw him, they were amazed and ran at once and greeted him. He asked them, "What are you arguing about with them?" A member of the crowd said to him, "Teacher, I brought you my son, who is possessed by a spirit that makes him mute. Whenever it seizes him, it throws him down, and he foams at the mouth, grinds his teeth, and becomes rigid. I asked your disciples to cast it out, but they were not able to do so." He answered them, "You unbelieving generation! How much longer must I be with you? How much longer must I endure you? Bring him to me." So they brought the boy to him. When the spirit saw him, it immediately threw the boy into a convulsion. He fell on the ground and rolled around, foaming at the mouth. Jesus asked his father, "How long has this been happening to him?" And he said, "From childhood. It has often thrown him into fire or water to destroy him. But if you are able to do anything, have compassion on us and help us." Then Jesus said to him, "'If you are able?' All things are possible for the one who believes." Immediately the father of the boy cried out and said, "I believe; help my unbelief!" Now when Jesus saw that a crowd was quickly gathering, he rebuked the unclean spirit, saying to it, "Mute and deaf spirit, I command you, come out of him and never enter him again." It shrieked, threw him into terrible convulsions, and came out. The boy looked so much like a corpse that many said, "He is dead!" But Jesus gently took his hand and raised him to his feet, and he stood up. Then, after he went into the house, his disciples asked him privately, "Why couldn't we cast it out?" He told them, "This kind can come out only by prayer." Mark 9:14-29 (NET)
In the morning as they passed by, they saw the fig tree withered from the roots. Peter remembered and said to him, "Rabbi, look! The fig tree you cursed has withered." Jesus said to them, "Have faith in God. I tell you the truth, if someone says to this mountain, 'Be lifted up and thrown into the sea,' and does not doubt in his heart but believes that what he says will happen, it will be done for him. For this reason I tell you, whatever you pray and ask for, believe that you have received it, and it will be yours. Whenever you stand praying, if you have anything against anyone, forgive him, so that your Father in heaven will also forgive you your sins." Mark 11:20-25 (NET)
Whose prayers move mountains?
Prayers of Elijah and other righteous men
Elijah's prayers changed the world (James 5:17-18). The prayers of righteous men still change the world (James 5:16, 19-20).
Prayers of the apostles and others who believe God's promises
Jesus told the apostles that they would receive whatever they asked in believing prayer, thereby doing works even greater than cursing the fig tree (Matt. 21:20-22). He told them they would have what they prayed for if they believed they would, and that God would forgive them if they forgave others as they prayed (Mark 11:24-25). That was not just for the apostles, but for all who heard the Sermon on the Mount (Matt. 6:14-15; 7:7-11; cf. Luke 11:5-13). If we don't believe we will have what we ask for, then we will not receive anything from the Lord (James 1:5-8).
Prayers of the apostles and others who believe in Jesus
God would give the apostles whatever they asked in the name of Jesus (John 15:22-24), as they remained in him (John 15:1-17). All those who believe will do greater works than Jesus did: Jesus will do whatever they ask in his name (John 14:12-14).
Through prayer, every believer does greater works than Christ did
"Most assuredly, I say to you, he who believes in Me, the works that I do he will do also; and greater works than these he will do, because I go to My Father. And whatever you ask in My name, that I will do, that the Father may be glorified in the Son. If you ask anything in My name, I will do it" (John 14:12-14, NKJV; cf. Matt. 21:20-22). Martin Luther characteristically took Jesus at his word:
Do not separate the head from its members, that is, Christ and his apostles and all Christendom. Every single Christian. accomplishes such great things that he can rule the whole world in divine matters, help and benefit everyone, and perform the greatest works that ever were effected on earth. For God thinks more of him than of the whole world; for his sake God gives and preserves to the world all it has. But if you are baptized, says Christ, and believe in me, you are the man who has and can do more and greater things, yea, who does the very works and greater ones than I do. For I shall make of you believers lords whose works shall count for more and accomplish more than those of any king or lord on earth; you shall bring about and achieve whatever you desire and shall help me rule spiritually over souls for their salvation, and also as to the material things you shall obtain through your prayer all that is on the earth. (St. L. VIII:350-356, quoted in Volume 3 of F. Pieper's Christian Dogmatics, pp. 80-81, emphasis added).
"Behold, I have given you authority to tread on serpents and scorpions, and over all the power of the enemy, and nothing shall hurt you. Nevertheless, do not rejoice in this, that the spirits are subject to you, but rejoice that your names are written in heaven" (Luke 10:19-20).

