Logarithm

In mathematics, the logarithm functions are the inverses of the exponential functions. If b>0 and x = b^y, we say that y is the logarithm of x in the base b (meaning y is the power we have to raise b to, in order to get x), and we write log_bx = y. For instance, log₁₀100 = 2 (since 10²=100) and log₂8 = 3 (since 2³=8).

Logarithms were invented by John Napier in the early 1600s. Before the widespread availability of electronic computers, logarithms were widely used as a calculating aid, both with tables of logarithms and slide rules. The basic idea here is that the logarithm of a product is the sum of the logarithms, and adding is easier than multiplying. In these applications, the base-10 or common logarithm was typically used.

Logarithms are also useful in order to solve equations in which the unknown appears in the exponent, and they often occur as the solution of differential equations because of their simple derivatives. Furthermore, various quantities in science are expressed by their logarithms; see logarithmic scale for an explanation and a list.

For integers a and b, the number log_ba, is irrational (equivalent to a decimal number with no repeating segment) whenever a and b rational numbers that are coprime (have no shared factor).

To calculate the derivative of a logarithm, the following formula is used

Where ln is the natural logarithm.

The function log_b(x) is defined whenever x is a positive real number and b is a positive real number different from 1. See logarithmic identities for several rules governing the logarithm functions.

There is a special base e (approximately 2.718) which has useful properties. The logarithm to this base is called the natural logarithm. When dealing with the logarithms to the base e, it is common especially to denote log_e by ln, especially if there is any likelihood that the reader might think that base 10 or base 2 logarithms might be meant. In most pure mathematical work, log or ln is used to denote log_e; in most engineering work, log means log₁₀; while in information theory, log often means log₂, which also sometimes is written as lg. Whenever a possibility for ambiguity exists, this ambiguity should be resolved by explicitly writing out the base.

To calculate the value for a logarithm of non base 10, the following formula will do a change of base so that a value may be obtained

where k is any valid base.

As mentioned, the base used extensively in information theory and computer science is the binary logarithm, base 2. It is used frequently because many algorithms and computer applications split items into two sub-items, in the divide-and-conquer manner. Binary logarithms are useful in determining characteristics of functions, such as the order of such functions that exhibit this behaviour.

A curious coincidence is the approximation log₂(x) ≈ log₁₀(x) + ln(x), accurate to about 99.4% or 2 significant digits.

The number, log_ba, is irrational (equivalent to a decimal number with no repeating segment) whenever the two numbers, a and b, are coprime (have no shared factor). (http://mathworld.wolfram.com/IrrationalNumber.html)

Logarithms may also be defined for complex arguments. This is explained on the natural logarithm page.

In the theory of finite groups there is a related notion of discrete logarithm. For some finite groups, it is believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications in cryptography.