Covariance of exponential X and lognormal Y given X

Let $X$ be exponential with rate $1$, and conditionally on $X=x$ let $\log Y$ be normal with mean $0$ and variance $x$. Compute $\operatorname{Cov}(X,Y)$.

$2$. Conditionally $\mathbb{E}[Y\mid X]=e^{X/2}$. Then $\mathbb{E}[Y]=\mathbb{E}[e^{X/2}]=2$ (the exponential MGF at $\tfrac12$) and $\mathbb{E}[XY]=\mathbb{E}[Xe^{X/2}]=\int_0^\infty xe^{-x/2}dx=4$. So the covariance is $4-1\cdot2=2$.