my question stems from this nice approximation of a percentage increase: for small percentage increases (usually below 20%), we can have an approximation with the natural log of the ratio. Here's an example:
50 increases to 55 --> % increase is (55-50)/50 (usually multiplied by 100, but let's keep it decimal for the time being)
OR
we could approximate it and say it is ln(55/50) or ln(55)-ln(50).
My question, then, is: is there a way to approximate a percentage decrease as well?
Think of it as a discount: the price was 55 and now it's 50. The discount is 1 - 50/55, or (55-50)/55. However, I really can't find a log approximation that simulates what the other one does and has the two following properties:
- The values of the percentage decrease and the approximation are similar for small percentage decreases.
- As the decrease gets larger (say, for instance, from 1000 to 50), the function (I'm assuming it'll be a log) keeps smoothing the decrease in a monotonic way (so from a linear enlargement of the decrease, we get a marginally diminishing one, just as it happens for the increase in the initial example).
Any ideas?
One thought I had was that, in general terms, you can positively express an increase as $(b-a)/a$ and a decrease as $(b-a)/b,$ so to get from the second to the first you can multiply the second by a/b. So you can find something decent with $\ln(b/a)*a/b$. If you do that, however, you end up with something like: $$ \begin{array}{lll} ~ & \text{% Decrease} & \text{Log decrease} \\ 50 & 0.0909 & 0.086645618 \\ 60 & 0.1667 & 0.151934631 \\ 100 & 0.5000 & 0.34657359 \\ 500 & 0.9000 & 0.230258509 \\ 1000 & 0.9500 & 0.149786614 \\ \end{array} $$