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:(adsbygoogle = window.adsbygoogle || []).push({});

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

Prove that (log_{10}a)/(log_{10}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 (log_{10}a)/(log_{10}b) is rational, then they are expressed as fraction:

(log_{10}a)/(log_{10}b) = p/q, where p and q are integers. q not 0. p not 0 because a is greater than 1.

Let 10^{x}= a

and 10^{y}= 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, 10^{kc}= a

and 10^{kz}= b

x = log_{10}a

y = log_{10}b

So, kc = log_{10}a

and kz = log_{10}b

But then, (10^{k})^{c}= a

and (10^{k})^{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 = log_{10}a

and y = log_{10}b

(log_{10}a)/(log_{10}b) is not rational.

Thank-you

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# Are my proofs 'correct'?

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