If $\log(a) - \log(b) \gt \log(c) - \log(d)$, must it always follow that $(a-b) \gt (c-d)$?
I'm looking over someone else's work, and they inexplicably took the logarithms of the numbers and then compared them, instead of just comparing the numbers themselves. He ends up throwing them into a sorted list and separates them out into quartiles for further analysis.
This got me to wondering, would the quartiles contain the exact same entities if he had never taken the logarithms in the first place?
Unfortunately, I can't see the underlying data, so I can't just figure it out for myself. Also, I'm just a law student checking a faculty member's work, so I want to make sure I know what I'm talking about before I suggest a change to how he describes his results.