Lognormal distribution inverse equivalent

Question

In Lognormal distribution if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. Is there inverse equivalent to lognormal distribution where Y = exp(X) has a normal distribution?

Answer 1

This is easily shown to be impossible for any real-valued random variable $X$: if we require $$Y = e^X \sim \operatorname{Normal}(\mu,\sigma^2),$$ then there must be some value of $X$ for which $e^X < 0$. But this is impossible if $X \in \mathbb R$.