Logarithm

In mathematics, a logarithm is a function that gives the in the equation bn = x. It is usually written as logb x = n. For example:

Using logarithms

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

Related Topics:
Positive - Real number - Logarithmic identities - Complex - Natural logarithm

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For integers b and x, the number logb(x) is irrational (i.e., not a quotient of two integers) if one of b and x has a prime factor which the other does not (and in particular if they are coprime and both greater than 1). In certain cases this fact can be proved very quickly: for example, if log23 were rational, we would have log23 = n/m for some positive integers n and m, thus implying 2n = 3m. But this last identity is impossible, since 2n is even and 3m is odd.

Related Topics:
Integer - Irrational - Prime factor - Coprime

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Unspecified bases

• Mathematicians generally understand either "ln(x)" or "log(x)" to mean log_e(x), i.e., the natural logarithm of x, and write "log₁₀(x)" if the base-10 logarithm of x is intended.
• Engineers, biologists, and some others write only "ln(x)" or (occasionally) "log_e(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log₁₀(x) or, in the context of computing, log₂(x).
• Sometimes Log(x) (capital L) is used to mean log₁₀(x), by those people who use log(x) with a lowercase l to mean log_e(x).
• The notation Log(x) is also used by mathematicians to denote the principal branch of the (natural) logarithm function.
• In most commonly used programming languages, including C, C++, Fortran, and BASIC, "log" or "LOG" means natural logarithm.

Most of the reason for thinking about base-10 logarithms became obsolete after the early 1970s when hand-held calculators became widespread (for more on this point, see common logarithm). Nonetheless, since calculators are made and often used by engineers, the conventions to which engineers were accustomed continued to be used on calculators, so now most non-mathematicians take "log(x)" to mean the base-10 logarithm of x and use only "ln(x)" to refer to the natural logarithm of x. As recently as 1984, Paul Halmos in his autobiography heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. (The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley.) As of 2005, some mathematicians have adopted the "ln" notation, but most use "log". In computer science, the base 2 logarithm is written as lg(x) to avoid confusion. This usage was suggested by Edward Reingold and popularized by Donald Knuth.

Related Topics:
Common logarithm - Paul Halmos - Berkeley - As of 2005 - Donald Knuth

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When "log" is written without a base (b missing from logb), the intent can usually be determined from context:

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• natural logarithm (log_e) in mathematical analysis;
• binary logarithm (log₂) with musical intervals and in subjects that deal with bits;
• common logarithm (log₁₀) when logarithm tables are used to simplify hand calculations;
• indefinite logarithm when the base is irrelevant.

Change of base

While there are several useful identities, the most important for calculator use lets one find logarithms with bases other than those built into the calculator (usually loge and log10). To find a logarithm with base b using any other base k:

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: log_b(x) = rac{log_k(x)}{log_k(b)}

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All this implies, moreover, that all logarithm functions (whatever the base) are similar to each other.

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