covariance of lognormal random variables I am trying to find the variance of b*log(x+y) - log(x), where x and y are independent and identically distributed lognormal random variables, the range for log(x) and log(y) is negative infinity to infinity, and 0<b<1. I have used the Fenton-Wilkinson (FW) approximation to find the variance of log(x+y), but I also need to find the covariance(log(x+y), log(x)) since both terms have x in common. FW doesn’t seem to help me with this problem (although I may be wrong). I had assumed that covariance(log(x+y), log(x)) = variance(log(x)) but my simulations do not support this answer.

This is really a comment that is too long for the comment box, but might spur someone else to prove the result.
There appears to be a very clean exact solution, namely that, if $X \sim \text{LogNormal}(\mu,\sigma)$ is independent of $Y \sim  \text{LogNormal}(\mu,\sigma)$, then:
$$\text{Cov}[\log(X+Y), \log(X)] = \frac12  \sigma^2$$
The proof, like Ramanujan, is ...
