Given two probability density functions (PDF) $f$ and $g$, both defined over the same set $X$, there are many ways to describe/measure the distance between them, e.g., KL divergence and Hellinger distance, to name a few.

**What I'm looking for** is a distance function $d(f,g)$ that gives *an order of magnitude difference*. To be concrete, if at some area $f\sim 2\times 10^4$, $f_1 \sim 10^4$, and $f_2 \sim 10^2$, I want that $d(f,f_1) \ll d(f,f_2)$. Is there a known, statistically or probabilistically meaningful way, to do that?

I know that this question is a bit vague, but if I knew how to define such a distance function, I wouldn't be asking this question. Note that, as in the examples above, $d(\cdot, \cdot)$ doesn't have to be a metric.

Thanks

*Edit*: This has been cross-posted in Cross.Validated