logarithm disambiguation

Unless otherwise specified, when I refer to the logarithm I am referring to the natural logarithm, and will denote it, as is customary in mathematics, by ‘log’. (Note that denoting the natural logarithm by ‘ln’ is an engineering convention, not a mathematical convention.)
Here is the Wikipedia article on the logarithm.


A logarithm is a certain type of injective (that is, invertible) function defined on the set of positive reals, whose importance is that it gives the option of accepting a small amount of error in exchange for a great increase in speed of computation. The logarithms match up exactly with the positive reals not equal to unity, the match-up number being called the logarithmic base of the given logarithm. The inverse of a given logarithm is called the exponential (with respect to the same base). The key property of a logarithm is that it converts products (which are difficult to compute) to sums (which are easy to compute). That is, if f is a logarithm, then for each positive real number x and for each positive number y, f(xy) = f(x) + f(y). It follows from this that if f is a logarithm, f(1) = 0, because f(1 times 1) = f(1) + f(1), and f(1 times 1) = f(1), and so f(1) = f(1) + f(1), and so f(1) = 0.

Logarithms can be distinguished not only by means of logarithmic base, but also by their derivative (that is, tangential slope) at unity. The logarithm having tangential slope of unity at unity is called the natural logarithm. The logarithmic base of the natural logarithm is denoted by ‘e’. The following estimate obtains:
2.71 < e < 2.72.

It turns out that all logarithms are proportional to one another. Therefore, the choice of logarithmic base is mathematically irrelevant, and is decided by convenience. The natural logarithm is most often convenient in theoretical mathematics, but the base-2 logarithm (aka the binary logarithm) is also sometimes convenient, and the base-10 logarithm has practical convenience.

keywords: mathematics

Skin in the game

“In any type of activity or business divorced from the direct filter of skin in the game, the great majority of people know the jargon, play the part, are intimate with the cosmetic details, but are clueless about the subject.”
— Nassim Taleb

A: “My friend William is thinking of getting married. What do you advise?”
B: “Yes, he should marry. If his wife is a good woman, he will be happy. If she is not a good woman, he will become a philosopher.”
(The relatedness to Taleb’s insight is that having a bad woman for a wife amounts to having ‘skin in the game’ in the game of life, and so the man will be far from clueless about life. In other words, he will be a philosopher.)

Let’s introduce a convenient handle for the type of person described by Taleb: A ‘skinless paragon’ is, by definition, any individual in a given activity or business divorced from the direct filter of skin in the game, but who knows the jargon, plays the part, is intimate with the cosmetic details, but who is clueless about the subject.

Now that we have a convenient handle for such a person, we can apply it. For example, database administrators are often clueless paragons, having been appointed to their position on account of their knowledge of the database system, not for their knowledge of the subject matter that the database deals with. Another example is the exposure of a skinless paragon in the Clint Eastwood movie ‘Trouble with the curve’. Another example is software / hardware developers: Initially they are in the same boat as end-users, and the features they implement are good ones. But as time goes by they have less and less knowledge of the corner conditions that actual end-users have to contend with, and the quality of the features they add suffers accordingly. For example, the end-key was placed at the end of the row, out of a naive match-up between the name of the key and its position on the keyboard, but this position is a very accessible one and should have been given to the home-key, because the home-key is used much more often. Another example is when an email application naively starts you in the ‘Recipients’ box because, after all that is the first box. However in order to guard against accidentally sending an incomplete message, or accidentally choosing the wrong recipient in my hurry to get to composing the message, I don’t fill this box in until the very last, and I’m sure that many others follow the same policy. The email application should start you in the message area, or, even better, allow you to configure where it starts you.

An old term for an especially blatant skinless paragon is ‘cracker-barrel philosopher’.

related term: ‘boots on the ground’

related term: ‘muscle memory’

related term: ‘hands-on experience’