Thinker8921
Aug23-11, 02:27 AM
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
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