This may be a naive question, but if I have random variables X and Y and take logs of both, would corr(log X, log Y) be greater than corr(X, Y)? Thank you in advance for your answer.
short answer: not necessarily. for example:
let X be a positive random variable with a pdf supported on some non-degenerate interval, [0, 1] say. let Y = 1 + X.
then X and Y are perfectly correlated. but U = logX and V = log Y = log (1 + X) = log(1 + e$^U$) are not linearly related, so their correlation is less than 1.
[X can also be discrete, as long as it assumes at least 3 different values with positive probability.]
this example can be tweaked so that X and Y start out not perfectly correlated, but still corr(U,V) < corr(X,Y).