Let us consider the following R.V. $S_1=S_0e^X_{1}$ and $S_2=S_1e^X_{2}$, where $X_1 \sim N(\mu_{1},\sigma_{1})$ and $X_2 \sim N(\mu_{2},\sigma_{2})$ are independen.$S_0$ is a constant. I know that the sum of two dependent normal is normal so that $S_2=S_0e^{X_1+X_2}$ is log normal. I think they are dependent, but how do I show it formally? For instance I may want to show that $f_{S_1S_2}(x,y) \neq f_{S_1}(x)f_{S_2}(y)$ or that $f_{S_2} \neq f_{S_2|S_1}$.
Edit: Is it helpful to down vote without elaborating?
Independence of $S_1$ and $S_2$ would imply that $\operatorname{Cov}(f(S_1),g(S_2))=0$ for any measurable functions $f$ and $g$ for which that covariance exists. In your case,
$$ \operatorname{Cov}(\ln(S_2),\ln(S_1))=\operatorname{Cov}(X_1,X_1+X_2)=\operatorname{Var}(X_1)\ne 0. $$