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In mathematics, the logarithm functions are the inverses of the exponential functions. If b>0 and x = by, 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 logbx = 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. Most people refer to the common logarithm when they speak of the logarithm.

The function logb(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 loge 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 loge; in most engineering work, log means log10; while in information theory, log often means log2, which is traditionally written as ld (from the Latin logarithmus dualis), but 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, assuming that a, b, and k are all positive real numbers and that and

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.

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 logba is irrational (i.e., not a quotient of two integers) if one of a and b has a prime factor which the other does not (and in particular if they are coprime and both greater than 1).

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

Where ln is the natural logarithm.

One can then see that the following formula gives the integral of a logarithm

A curious coincidence is the approximation log2(x) ≈ log10(x) + ln(x), accurate to about 99.4% or 2 significant digits. Another interesting coincidence is that, approximately, log102=0.3 (the actual value is about 0.301029995). This is ultimately due to the fact that, to within 3%, 210=103 (ie 1024 is about 1000).

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.