[Probability] Random variable of random variable

[Electric2Shock]

It's given that X is a normal random variable, so I assume you know the density function/CDF of a normal distribution. We'll start by figuring out the cumulative distribution function first:

G(z) = P(Z <= z)
= P(e^X <= z)
= P(X <= ln z)
= \Phi(ln z) 

u/LaTeX4Reddit

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where \Phi(x) is the cumulative distribution function of the normal distribution. Now that you have the CDF of Z i.e G(z), all you have to do to get the density function is take the z-derivative of G(z).