Log-scale informs on relative changes (multiplicative), while linear-scale informs on absolute changes (additive). When do you use each? When you care about relative changes, use the log-scale; when you care about absolute changes, use linear-scale. This is true for distributions, but also for any quantity or changes in quantities.
Say you have a function $xy=1$ with doubly linear scale. You convert the scale to a doubly exponential scale. What I mean by a "doubly exponential scale" is that it is the inverse of the log-log scale.
What does a doubly exponential scale inform on?
In general when you convert the scale to exponential what is preserved with regards to the function after the conversion?
Log and log-log graphs is used to fit rapidly increasing data onto a page. Exp and exp-exp graphs would be used to fit rapidly decreasing data onto a page without cramming all the small stuff so close together that the data blurs.