# Variance of the logarithm of the product of two random variables

I need to calculate the variance of the following product on the log scale, $$Z \sim \alpha \beta X Y$$ where $\alpha$ and $\beta$ are both scalars and $X \sim LN(\mu, \sigma^2)$ and $Y \sim U(a, b)$. $X$ and $Y$ are independent of each other. What I'm interested in is, $$Var(\log Z)$$ On the log scale the scalars will not affect the variance, so is this correct? $$Var(\log Z) = Var(\log(XY)) = Var(\log(X) + \log(Y)) = Var(\log X) + Var(\log Y)$$ I know $\sigma^2$, which is $Var(\log X)$, and I can calculate $Var(\log Y)$ form here.

Is this approach valid?

• You may need to assume that parameter $a>0$ (currently not stated). $$Var(\log Y) = 1-\frac{a b (\log (b)-\log (a))^2}{(b-a)^2}$$ – wolfies Jul 18 '17 at 16:35
• You are correct, that's the way to go. – jbowman Jul 18 '17 at 17:09