Distribution of $\log(x)$ for exponentially distributed $x$.

Question

If a dataset is exponentially distributed, i.e. we have a PDF=$\lambda \exp(-\lambda x)$ and CDF=$1-\exp(-\lambda x)$, then what is the distribution of $y=\log(x)$?

The reason I am interested in this is that fitting the exponential distribution may be fairly insensitive to what happens at really low $x$. Fitting the distribution of $\log(x)$, I believe, will help recover the low $x$ tail.

Answer 1

Let $Y = \ln X$. The cdf for $Y$ is $$G(y) = P (Y \le y ) = P (\ln X \le y ) = P (X \le e^y) = F (e^y) = 1 - e^{-\lambda e^{y}}$$