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?