correlation question

Dear R-helpers,

First, double apologies because the question is not
directly related to R, and also probably quite simple.

I have observed (with many data) that given 2 series X and Y
cor(X,Y) and cor(log(X), log(Y)) differs only very slightly.
I suspect this is because log(x) is monotonous,
... but I am unable to demonstrate it.
(perhaps it is also wrong ?).

Would somebody here be able and kind enough to put me on
the rigth track (a piece of clue or a link would be nice).

Thanks
Vincent