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Prove that Log A is Irrational


Date: 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/   
    
Associated Topics:
High School Exponents
High School Number Theory

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