I have a log scaled volume density distribution, $q_{3,log}$ from which I want to get number fraction, $\Delta Q_0$ with normal scale. So to transform $q_{3,log}$ to $\Delta Q_3$ the used relation is $\Delta Q_3 = q_{3,log}*log(x_i/x_{i-1})$ as indicated here.
But for the further transform of $\Delta Q_3$ to $\Delta Q_0$ with normal scale, to my opinion I should apply $\Delta Q_{0,i} = \Delta Q_{3,i} /x_i^3$, I find entire number fraction is located in the first few bins. Later, if I apply $\Delta Q_{0,i} = \Delta Q_{3,i} /log(x_i^3)$ I get reasonably "better-looking" distribution, but I think it is mathematically incorrect. So, I want opinion of mathematicians about what is correct approach and how to deal distribution when converting log-scaled volume density $q_{3,log}$ to number fraction, $\Delta Q_0$ with normal scale. How to get most reasonable number distribution from volume distribution?
