Why would you multiply a function back by its natural log? Or do something in the form of [x ln(x)] / [y ln(y)]?
3 Answers
For natural numbers n, n ln n is roughly the logarithm of the factorial n! which makes it relevant in many combinatorial settings (for more, see the Stirling approximation, e.g., http://en.wikipedia.org/w
The most common application arises in computer science, where we can show that the best sorting algorithms cannot have an average case performance better than O(n ln n) (computer science uses binary rather than natural logarithms, but this is covered in the big-oh notation).
There are probably better references, but here's a starting point for sorting algorithms: the Wikipedia page http://en.wikipedia.org/w
The most common application arises in computer science, where we can show that the best sorting algorithms cannot have an average case performance better than O(n ln n) (computer science uses binary rather than natural logarithms, but this is covered in the big-oh notation).
There are probably better references, but here's a starting point for sorting algorithms: the Wikipedia page http://en.wikipedia.org/w
One reason would be in order to measure entropy, so x or y has to be a probability. -xlnx is then the entropy. Or, I should say entropy is the negative sum over all your possible x's of that value. So if the random variable X can equal 0 (tails) or 1 (heads) with .5 probability each, the total entropy is calculated as -(.5*ln(.5)+.5*ln(.5)), which is about 0.693.
http://en.wikipedia.org/w
A neat thing I read about this definition recently:
http://johncarlosbaez.wor
Not totally relevant, but this definition also reminds me of pH because the definition of pH is a negative of a logarithm as well; it's the negative logarithm of a concentration. Whenever people take the logarithm of a number they know is between zero and one, they tend to flip it and make it positive!
http://en.wikipedia.org/w
A neat thing I read about this definition recently:
http://johncarlosbaez.wor
Not totally relevant, but this definition also reminds me of pH because the definition of pH is a negative of a logarithm as well; it's the negative logarithm of a concentration. Whenever people take the logarithm of a number they know is between zero and one, they tend to flip it and make it positive!
