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.

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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