Determine Normal Distribution based on LogNormal Distribution

Question

Suppose that we have that $Y=e^{aX}$ where $a$ is a positive scalar. We know that $Y$ follows a logNormal distribution with parameters $0$ and $2$. Then is there a way to derive the distribution of $X$??

From the definition, I think the $log(Y)=aX$ has to follow a normal distribution with mean $0$ and standard error $2$, but this distribution corresponds to $aX$. Thus, I would like to find a distribution of just $X$.

Something that came to my mind right now is that $aX$ (this kind of transformation) results to changes in the variance of the Normal distribution.

Answer 1

If $Y=e^{aX}$ is log-normal with log-mean $0$ and log-variance $2$, then $aX$ is normal with mean $0$ and variance $2$.

Rescale $aX$ by $\frac{1}{a}$, then $X=\frac{1}{a}\times aX$ is normal with mean $\frac{0}{a}=0$ and variance $\frac{2}{a^2}$.

If the parameters you refer to are the log-mean and the log-standard deviation instead of the variance, then the standard deviation of $X$ will be $\frac{2}{|a|}$.