Don't know if anybody is still following this thread but to clarify ...
The MOND "formula" is not really a true formula and the MOND "theory" is not a true theory. The theory is really just a suggestion that the Newtonion formula may not be exactly correct and suggests what the true formula...
You are absolutely right. The non-linearity can create these oddball situations. When I work these problems out I normally think of the correlation as distinguishing between a single correlated component and a single uncorrelated component. For linear operations this approach is valid. But when...
Well, not "precisely". The reality is that it is not undefined. It is just not real. E.g.
ln(-1) = i*3.1415927
Similarly the correlation is not undefined although it could perhaps have imaginary components.
In other words, the question is not moot it is just "complex" (pun intended).
Of course, that's what I said. But that doesn't prove that there isn't a direct, even linear, relationship between the variable correlations and the log correlations. Seems unlikely but as yet I haven't found a proof one way or the other.
Well, by that argument log of normal is undefined...
Actually you're right. I'm not thinking clearly.
This one you're not thinking clearly about. If x and y are uncorrelated then it is not possible that their logs could have any correlation. The log transformation does not introduce any component that they could have in common.
If they are...
Well, there are all sorts of approximations I can devise but they would not be precise. Since log(x) and log(y) are both normal (i.e. log of normal is still normal) there should be a simple correlation coefficient that can be calculated and used. I presume, then, there should be a closed form...
I was trying to build a probability-related software package and needed to have a theoretical framework to deal with some less common issues (i.e. stuff that you don't find in the average textbook). I was hoping that somebody could give me pointers as to where to find the proper formulae...