# Are my proofs 'correct'?

1. Aug 23, 2011

### Thinker8921

I can never seem to create proofs the way it is shown in every textbook I've seen. To be honest, I don't really know how to write the proofs correctly. I've seen sometimes my reasons are flawed and other times I go around aimlessly and get home after some unnecessary steps. So I would just like some feedback here:

1. The problem statement, all variables and given/known data
Prove that (log10 a)/(log10 b) is not rational if a and b are relatively prime and both are greater than 1.

2. Relevant equations

3. The attempt at a solution
Suppose that (log10 a)/(log10 b) is rational, then they are expressed as fraction:
(log10 a)/(log10 b) = p/q, where p and q are integers. q not 0. p not 0 because a is greater than 1.
Let 10x = a
and 10y = b
x and y cannot be zero because a and b cannot be 1.
Now if a and b are relatively prime, ie they have only 1 as their common factor then x and y too are relatively prime.

Reason:
Suppose that x and y share a common factor greater than 1, then:
let x = kc
and y = kz
where k is the common factor, which is not negative or 0 because a and b are greater than 1 and c and z are also not 0 or negative.
So then, 10kc = a
and 10kz = b
x = log10 a
y = log10 b
So, kc = log10 a
and kz = log10 b
But then, (10k)c = a
and (10k)z = b
And this contradicts the fact that a and b are relatively prime. So x and y are also relatively prime.

So now we can write:
x/y = p/q
i.e, xq = py
This cannot be true because x and y are relatively prime. This means x/y is not rational.
And since x = log10 a
and y = log10 b
(log10 a)/(log10 b) is not rational.

Thank-you

Last edited: Aug 23, 2011
2. Aug 23, 2011

### tiny-tim

Hi Thinker8921!

hmm … this is really complicated

you're introducing x and y for no particular reason.

Just do …
… then write that as q(log10 a) = p(log10 b), and carry on from there.

(or better still use logb a, if you know how to do that)