Re: LOG(3) / LOG(2)
- From: Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Sun, 13 Jan 2008 23:19:53 GMT
In article
<2ee5dccb-b13c-44dd-8c9a-03cf46cb8ff5@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Dave <dave_and_darla@xxxxxxxx> wrote:
On Jan 12, 6:00 pm, Gerry Myerson <ge...@xxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:
In article
<0f142fda-1417-4631-9738-ca134b010...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Dave <dave and da...@xxxxxxxx> wrote:
On Jan 12, 2:52 pm, bill <b92...@xxxxxxxxx> wrote:
Prove that LOG (3) / LOG(2) is irrational.
Proceeding indirectly, assume that log(3)/log(2) is rational, i.e.,
let log(3)/log(2) = p/q with p and q integers and q>0.
Then, p log(2) = q log(3)
So log(2^p) = log(3^q)
So 2^p = 3^q.
But this is impossible because the left side is even
2^0 is even? 2^(-17) is even?
Be a bit more careful with your hypotheses.
Didn't you see my hypothesis that q>0? Of course, that implies that
p>0 also since both log(2) and log(3) are positive.
Then it's a question of what you put into the proof explicitly,
and what you leave to the reader to supply for herself. I figure
that someone who can't prove on her own that log 3 / log 2 is
irrational may not notice the implication p > 0, so I'd put it in
and not leave anything to chance.
--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.
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