I am totally blind to this verification, is this true?
$\operatorname{corr}(X,Y)=\operatorname{corr}(log(X),log(Y))$
$$ \operatorname{corr}(X,Y)=\frac{\operatorname{cov}(X,Y)}{\operatorname{var}(X)\operatorname{var}(Y)}\\ \operatorname{corr}(ln(X),ln(Y))=\frac{\operatorname{cov}(ln(X),ln(Y))}{\operatorname{var}(ln(X))\operatorname{var}(ln(Y))} $$
Thank you for any comments
Even writing out the definition of each, I have no idea how to compare them.
Such a claim cannot possibly be true; e.g., if $\Pr[X < 0] > 0$ or $\Pr[Y < 0] > 0$, the RHS is undefined but the LHS still exists.
Even if you restrict the support of $X$ and $Y$ to be positive, then even a simple location-transformed Bernoulli example will show otherwise; e.g., $$\Pr[(X,Y) = (1,1)] = 1/6 \\ \Pr[(X,Y) = (2,1)] = 1/3 \\ \Pr[(X,Y) = (1,4)] = 1/8 \\ \Pr[(X,Y) = (2,4)] = 3/8$$ gives $$\rho_{XY} = \frac{25}{3\sqrt{383}}$$ but $$\rho_{\log X \log Y} = \frac{1}{\sqrt{429}}.$$