# Is there a name for a distribution where I can take log of its histogram and get back the same histogram?

I am imagining some data distribution where taking the log over and over again yields a distribution that has the exact same shape.

• Please explain what you mean by "take log of [a] histogram."
– whuber
Oct 5, 2022 at 14:54

To put this in more mathematical form, you want to be able to start with a random variable $$X$$, take a logarithm, then perhaps add and multiply by some numbers and get back the same distribution. $$Y=a+b\log X$$ and $$X$$ have the same distribution.
It's not possible otherwise. Let $$x_1$$ and $$x_2$$ be two values of $$X$$ and consider the point halfway between them: $$x_m=\frac{1}{2}(x_1+x_2)$$. Because $$\log$$ is a concave function, $$\log x_m$$ will always be less than $$\frac{1}{2}(\log x_1+\log x_2)$$. If we write $$y_1$$ and $$y_2$$ for the transformed values of $$x_1$$ and $$x_2$$, we can choose the scaling so that $$y_1=x_1$$ and $$y_2=x_2$$. But $$y_m, rather than $$y_m=x_m$$. So we can only get the same shape back if there is no probability at $$x_m$$. Basically the same argument works for a point 1/3 of the way between $$x_1$$ and $$x_2$$, or any other intermediate position.