# Covariance of natural logarithms — how to estimate $\sigma (\mathbf X, \mathbf Y)$ from $\sigma (\ln{\mathbf {(X)}},\ln{\mathbf {(Y)}})$?

I have

$$\operatorname{Cov} (f{\mathbf {(X)}},f{\mathbf {(Y)}})$$

where $\operatorname{Cov}$ denotes the covariance and $f(\mathbf X)$ is a nonlinear function, i.e. $f(\mathbf X) = \ln(\mathbf X)$. Is there some transformation to estimate the covariance of the underlying variables? Such as:

$$\operatorname{Cov} (\mathbf X, \mathbf Y)$$

The background: I am estimating the expected error between two datasets. One has an error covariance matrix $\mathbf{S_1} = \operatorname{Cov}(\mathbf X,\mathbf Y)$, the other has an error covariance matrix $\mathbf{S_2} = \operatorname{Cov}(\ln \mathbf X, \ln \mathbf Y)$. To estimate the covariance for the comparison ensemble, I need to somehow add those two up; but how do I convert the covariance of the natural logarithms to the covariance of the quantity itself?

More directly, using the results derived there, if $U=\log(X)$ and $V=\log(Y)$ ($\log$ being the natural log, naturally), the analysis in that question gets us to (after some rearrangement):
$$\operatorname{Cov}(X,Y) \approx \operatorname{E}(X)\operatorname{E}(Y)\left(e^{\operatorname{Cov}(U,V)}-1\right)\,.$$
This approximation is exact for $U,V$ bivariate normal. If the bivariate distribution on the log-scale is reasonably close to normal it should work quite well, especially in large samples.