Question Concerning the Invariance of a Log-Transformed Normal Random Variable under Reciprocal Transformations So I just started looking through through E.J. Gumbel's "Statistics of Extremes" (1958) and I came across a rather strange problem that I had never seen before. The problem is phrased as follows:

(1) Show that the distribution obtained from a logarithmic transformation of a normal variate is invariant under a reciprocal transformation. (2) How is the mode of the transformed distribution related to the transformation of the initial mode?

I could understand going in the direction of
$Y = e^{X}$,
where  $X \sim N(\mu,\sigma)$, to obtain the pdf of the log-normal distribution (since there is at least the special case of $\mu = 0$ where it is indeed invariant under reciprocal transformations); however, it seems like Gumbel wants the reader to work with
$ Y = \log{X}$,
where $X \sim N(\mu, \sigma)$, to obtain the pdf of the resulting distribution and then obtain the pdf of
$W = Y^{-1}$
to show that we still obtain the same pdf as before. My initial thought was "wouldn't I need to do some kind of truncation to restrict X for log(X) to make sense?", but I figured that maybe Gumbel was just assuming that I would know this, so I went ahead and just assumed that we are only dealing with a truncated normal distribution that bounds X below by 0 but allows it to be arbitrarily large. The answers I'm getting don't seem to reward me with "invariance under reciprocal transformations" (I can't tell if he means that we are staying within the same family of distributions or whether I am literally still getting the exact same distribution).
Anyways, Gumbel has a tendency to use different terms for familiar concepts, so it may be that I am misunderstanding what he asks; otherwise, how is this question not fundamentally non-sensical? If I am missing something really obvious, then I would happy with any advice that I can get.
P.S.: I am aware of other books on extreme values. I just wanted to try this specific textbook out.

After reading further into the book, I am able to confirm that @whuber is right when they say that Gumbel meant $Y = exp(X)$ and $X = log(Y)$. Here is the evidence from chapter 1.1.9. The Logarithmic Normal Distribution which is on page 16 of the 2004 Dover version of the text.

The distribution which obtained from the normal one by a logarithmic transformation has often been used in problems connected with extremes.

He then goes on to show the form of this distribution

Let f(y) and F(y) be the normal functions. Then the distribution g(z) and the probability G(z) obtained from the transformation $y = lg\ z$ are $g(z)=\frac{1}{z \sqrt{2 \pi}} exp(-\frac{lg^{2}z}{2});\ G(z) = F(lg\ z).$

which is the pdf for the lognormal distribution. The wording from the first quote is the exact wording that he used to describe the variate in the original problem. I'm posting this so that those who explore the book in the future will be aware that the question is referring to a lognormal distribution.
