# Proof that Log2 of 5 is irrational

1. Nov 24, 2007

### Pascal's Pal

1. The problem statement, all variables and given/known data

Prove that log2 of 5 is irrational.

2. Relevant equations

None.

3. The attempt at a solution

I just had a glimpse of the actual solution, but I'm wondering if mine would work too.

2^(a/b) = 5

square both sides...

2^(2a/b) =25

2 = 25^(b/2a)

(b/2a) = log25 of 2

b = 2aLog25 of 2

b is even...

and through a similar process...by taking the square root of both sides of "2^(a/b) = 5" you can arrive at a being even too. So how can they both be even etc etc.

2. Nov 24, 2007

### Kummer

$$\log_2 5 = a/b$$ so $$2^{a/b} = 5 \implies 2^a = 5^b$$. Now use unique factorization.

3. Nov 24, 2007

### Pascal's Pal

But does mine work?

4. Nov 24, 2007

### rock.freak667

What is unique factorization ?