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The Latter-day Saints make the claim to have restored two priesthoods: one deemed Aaronic, the other after the order of Melchizedek. Little if any of the justification for this comes from the Biblical text -- the supposed restoration of the Priesthoods of Aaron and Melchizedek was said to have come as a result of Joseph Smith and Oliver Cowdrey receiving successive visitations from Peter, James and John [Talmage, Articles of Faith, 170-1; for a description of these, see ibid., 186-7; see also Griffith, One Lord, One Faith, 111]. It is the Mormon claim that these priesthoods existed in the early church as well [Griffith, 113].
As our main concern has been with Mormon use of the Bible, and as the bestowal of these priesthoods rests upon the supposition of Mormon prophetic authority, our treatment of this subject will be very brief.
Concerning the Aaronic priesthood, Mormon apologists may note that this priesthood is an "everlasting" one (Numbers 25:10-13; Exodus 40:15). The word used in these verses for "everlasting" is the Hebrew 'olam. Barr discusses this word extensively in his work Biblical Words for Time, and translates 'olam as meaning "in perpetuity."
Does this mean, though, that the thing described will last forever? Probably not. One may argue that the thing described is intended to last forever, but 'olam is clearly used in places to describe something which will end:
Exodus 21:6 Then his master shall bring him unto the judges; he shall also bring him to the door, or unto the door post; and his master shall bore his ear through with an awl; and he shall serve him for ever.
There is an obvious limitation involved here, namely, the life of the slave (cf. 1 Sam. 1:22-8). Here, then, 'olam stresses permanence, and assumably, the man would be a slave forever if he lived that long. But 'olam is used not of things that "will" end, but things that did end, but were meant not to. Specifically, it is used to ordinances in the Jewish law which were to be kept by the Israelites. The covenant and priesthood was intended to last forever, but obviously did not.
Nevertheless Griffith  offers the argument that there is no indication that the Aaronic priesthood was ever abolished. Perhaps not in so many words -- but the OT clearly indicates that membership in this priesthood was limited to descendants of Aaron. Of course we may expect it to be argued that Smith's revelation was an update on this point -- and at such point, the argument turns on the assumption of Mormon prophetic authority and goes beyond our scope.
What, then, of Melchizedek's priesthood? A key verse here is Hebrews 7:24:
But this man, because he continueth ever, hath an unchangeable priesthood.
"This man" refers to Christ, and Christian apologists argue that the word "unchangeable" carries the meaning "untransferable" so that any claim by Smith to have acquired the rights, so to speak, to this priesthood are in direct opposition to Hebrews 7:24. Griffith  cites "the best Greek scholars" -- including Leon Morris and W. E. Vine -- who says that this interpretation is not justified.
Perhaps not -- but the context indicates that the intent in saying that the priesthood is "unchangeable" is in effect declaring that Jesus is the last of the line. 7:23 says, "And they truly were many priests, because they were not suffered to continue by reason of death..." Verse 16 refers to Christ's "endless life." The high priest served until death under ordinary circumstances; under Rome he might serve till removal, but either way no priest could serve eternally -- until Jesus. That's because he won't die, and no one will remove him. Therefore, even allowing the "unchangeable" interpretation, the argument that Smith was in conflict with Hebrews 7:24 in claiming the Melchizedek priesthood remains intact.
He could not claim it until such time as Jesus died or he managed to kick him out of the office -- and no one thinks that that has or will happened in any event.