# Determine Normal Distribution based on LogNormal Distribution

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.

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 $$\frac{2}{|a|}$$.