if the distribution of a positive random variable $X$ form a scale family, how can I show the distribution of $LogX$ form a location family?

Question

if the distribution of a positive random variable $X$ form a scale family, how can I show the distribution of $LogX$ form a location family?

It's obviously true, but I have no idea how to prove it, any help will be appreciated.

Answer 1

Let $\log(X)$ have distribution function $G$, and let $\log(aX)$ have distribution function $H$. Then $$G(x) = P(\log(X)\leq x) = P(\log(a)+\log(X)\leq \log(a)+x) = P(\log(aX)\leq \log(a)+x) = H(\log(a)+x).$$

The distribution functions of $\log(X)$ and $\log(aX)$ are thus shifted versions of each other, so $\log(aX)$ form a location family.