Edit 10/25/2019:
$f(x)$ could also be made a probability distribution by introducing a parameter and normalizing constant with $K_1$ being a modified bessel function of the second kind
$$f(x)=\frac{1}{2\sqrt{{S}}~K_1\big(2\sqrt{S}\big)}~e^\frac{{\large S}}{\ln(x)}$$
for $\Re(s)>0,$ and in this case the expected value is well-defined
$$E[X]=\int_0^1 xf(x)~dx=\frac{K_1(2\sqrt{2}\sqrt{S})}{\sqrt{2}K_1(2\sqrt{S})} $$
After solving $f(\frac{1}{x})=\frac{1}{f(x)}$ I took one of the solutions, $f(x)=\exp\bigg(\frac{1}{\ln(x)}\bigg),$ found $f'(x)$ for the reciprocal of $f(x),$ and thought that maybe it could be made into a probability distribution. I chose the normalizing constant $\rho$ and found that
$$\nu(x)=\frac{\rho\exp\bigg(-\frac{\rho}{\ln(x)}\bigg)}{x\ln^2(x)}$$
is a probability mass function for $\Re(\rho)\ge1.$
This is because $$V(x)=\int_1^\infty \nu(x)~~dx=1$$ for all $\Re(\rho)\ge1.$
I noticed $$E[X]=\int_1^\infty x~\nu(x)~~dx $$ is undefined.
Q: Do these probability distributions have any applications? What are their entropies?