Prove that Log A is IrrationalDate: 06/14/98 at 20:44:39 From: Magictek Subject: Prove that log A is irrational - {A| A<>10^n} Can you help me prove that a common log of a number (not powers of 10) is irrational? For example log 3, log 4 , etc... I can only go up to A=10^N where N is the irrational number, and I think I have to prove that all values of N will not make A unless it is irrational. Date: 06/16/98 at 22:50:36 From: Doctor Schwa Subject: Re: Prove that log A is irrational - {A| A<>10^n} Generally the way to prove a number is irrational is to suppose that it's rational, and then get a contradiction, showing your supposition must be impossible. Suppose A = 10^(x/y). Then A^y = 10^x. The key is to recognize that you can reason in terms of prime numbers here, 10 = 2*5, so 10^x = 2^x * 5^x. Since every whole number has a unique prime factorization, A^y = 2^x 5^x also. How can that happen? Well, the only primes that go into A must be 2 and 5, so A = 2^a 5^b, and thus A^y = 2^(ay) 5^(by). And then ay = x, and by = x also, so a = b, and thus A is a perfect power of 10. -Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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