# Logarithm

In mathematics, a logarithm of x with base b may be defined as the following: for the equation bn = x, the logarithm is a function which gives n. This function is written as n = logb x. Logarithms tell how many times a number x must be divided by the base b to get 1, and hence can be considered an inverse of exponentiation. For example:

Missing image
Logarithms.png
Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1.7. Note how all basic logarithms pass through the point (1, 0).

[itex]\log_3(81) = 4[itex], because [itex]3^4 = 81[itex]

Logarithms are one of three functions that can be used to solve the equation bn = x for any variable, given the other two. The others are the radical, which can be used to find b (b is the nth root of x), and the exponential function, which can be used to find x (x is the nth power of b).

The logarithm functions are the inverses of the exponential functions. Logarithms convert multiplication to addition, division to subtraction (making them isomorphisms between the field operations), exponentiation to multiplication, and roots to division (making logarithms crucial to slide rule construction).

A negative logarithm has a negative sign. This function is written as n = - logb x. It is used for example in chemistry to express the concentration of protons (pH).

An antilogarithm is used to show the inverse of the logarithm. It is written antilogb(n) and means the same as bn.

A double logarithm is the inverse function of the double-exponential function. A super-logarithm or hyper-logarithm is the inverse function of the super-exponential function. The super-logarithm of x grows even slower than the double logarithm for large x.

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.

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## Use

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 are 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 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.

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

## Base systems

When logarithms are repeated in series, usually only one base (b in bn = x) is defined, and afterward log(x) can be shorthand for the longer form logb(x). So, in a system of logarithms of which 8 is declared the base:

 log(8) = 1 antilog(1) = 8 log(64) = 2 antilog(2) = 64 log(512) = 3 antilog(3) = 512 log(4096) = 4 antilog(4) = 4096 ...

There are three widely used bases, so those logarithms are given their own names: binary (base 2), natural (base e), and common (base 10) logarithms.

### Change of base

One's choice of base with logarithms is not crucial, because a logarithm can be converted from one base to another quite easily. For example, to calculate the value of a logarithm of a base other than 10, given a table or calculator that can only handle base 10, the following formula changes the base to any chosen base (assuming that b, x, and k are all positive real numbers and that b ≠ 1 and k ≠ 1)

[itex]\log_b(x) = \frac{\log_k(x)}{\log_k(b)}[itex]

where k is any valid base. Letting k = x gives

[itex]\log_b(x)=\frac{1}{\log_x(b)}[itex]

To see why this is the case, consider the following equations:

 [itex]b^{\log_b(x)} = x\!\,[itex] by definition [itex]\log_k\left( b^{\log_b(x)} \right) = \log_k(x)[itex] take logs on both sides [itex]\log_b(x)\, \log_k(b) = \log_k(x)[itex] simplify the left hand side [itex]\log_b(x) = \frac{\log_k(x)}{\log_k(b)}\,\![itex] divide by logk(b)

Furthermore, in some applications, one can avoid choosing any particular base at all by instead just using the indefinite logarithm.

### Relationships between binary, natural and common logarithms

In particular we have:

log2(e) ≈ 1.44269504
log2(10) ≈ 3.32192809
loge(10) ≈ 2.30258509
loge(2) ≈ 0.693147181
log10(2) ≈ 0.301029996
log10(e) ≈ 0.434294482

A curious coincidence is the approximation log2(x) ≈ log10(x) + ln(x), accurate to about 99.4% or 2 significant digits; this is because 1/ln(2)1/ln(10) ≈ 1 (in fact 1.0084...). The property is demonstrated in all six conversion factors above, arranged in pairs of two:

 2.3 3.32 0.3 0.69 0.43 1.44

This comes on top of the reciprocal relations we have:

 2.3 0.43 0.3 3.32 0.69 1.44

Another interesting coincidence is that log10(2) ≈ 0.3 (the actual value is about 0.301029995); this corresponds to the fact that, with an error of only 2.4%, 210 ≈ 103 (i.e. 1024 is about 1000; see also Binary prefix).

## Applications in calculus

To calculate the derivative of a logarithmic function, the following formula is used

[itex]\frac{d}{dx} \log_b(x) = \frac{1}{x \ln(b)} = \frac{\log_b(e)}{x}[itex]

where ln is the natural logarithm, i.e. with base e. Letting b = e :

[itex]\frac{d}{dx} \ln(x) = \frac{1}{x}, \qquad \int \frac{1}{x} \,dx = \ln(x) + C[itex]

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

[itex]\int \log_b(x) \,dx = x \log_b(x) - \frac{x}{\ln(b)} + C = x \log_b \left(\frac{x}{e}\right) + C[itex]

## Algorithm

To calculate logb(x) if b and x are rational numbers and xb > 1 :

If n0 is the largest natural number such that bn0x or, alternately,

[itex] n_0 = \lfloor \log_b(x) \rfloor [itex]

then

[itex] \log_b(x) = n_0 + \frac{1}{\log_{x / b^{n_0}}(b)} [itex]

This algorithm recursively produces the continued fraction

[itex] \log_b(x) = n_0 + \frac{1}{n_1 + \frac{1}{n_2 + \frac{1}{n_3 + \cdots}}}. [itex]

The logarithms produced are irrational for most inputs.

This algorithm works because :

 [itex] \log_b(x) = \log_b(x)\,\! [itex] identity [itex] \log_b(x) = n_0 + \log_b(x) - n_0\,\! [itex] algebraic manipulation [itex] \log_b(x) = n_0 + \log_b(x) - \log_b(b^{n_0})\,\! [itex] logarithmic identity [itex] \log_b(x) = n_0 + \log_b\left(\begin{matrix}\frac{x}{b^{n_0}}\end{matrix}\right) [itex] logarithmic identity [itex] \log_b(x) = n_0 + \frac{1}{\log_{\begin{matrix}\frac{x}{b^{n_0}}\end{matrix}}(b)} [itex] base switch

To use irrational numbers as inputs, apply the algorithm to successively detailed rational approximations. The limit of the result sequence should converge to the actual result.

[itex]\ln 2 = 2\sum_{n = 0}^{\infty }\frac{(1/5)^{

2\,n+1}}{ 2\,n+1}+2\sum_{n = 0}^{\infty }\frac{(1/7)^{ 2\,n+1}}{ 2\,n+1}[itex]

## History

Joost Bürgi, a Swiss clockmaker in the employ of the Duke of Hesse-Kassel, first conceived of logarithms. The method of natural logarithms was first propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier (c. 1550 - 1618; Latinized Neperus), Baron of Merchiston in Scotland, four years after the publication of his memorable invention. This method contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, it was constantly used in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved prosthaphaeresis, which relied on trigonometric identities, as a quick method of computing products. Besides their usefulness in computation, logarithms also fill an important place in the higher theoretical mathematics.

At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word logarithm, a portmanteau, to mean a number that indicates a ratio: λoγoς (logos) meaning ratio, and αριθμoς (arithmos) meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers for which they stand, so that an arithmetic series of logarithms corresponds to a geometric series of numbers. The term antilogarithm was introduced in the 1800s and, while convenient, its use was never widespread.

### Tables of logarithms

Prior to the advent of computers and calculators, using logarithms meant using tables of logarithms, which had to be created manually. Base-10 logarithms are useful in computations when electronic means are not available. See common logarithm for details, including the use of characteristics and mantissas of common (i.e., base-10) logarithms.

In 1617, Briggs published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal places. This he followed, in 1624, by his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals, together with a learned introduction, in which the theory and use of logarithms are fully developed. The interval from 20,000 to 90,000 was filled up by Adrian Vlacq, a Dutch computer; but in his table, which appeared in 1628, the logarithms were given to only ten places of decimals.

Vlacq's table was later to found to contain 603 errors, but "this cannot be regarded as a great number, when it is considered that the table was the result of an original calculation, and that more than 2,100,000 printed figures are liable to error." (Athenaeum, 15 June 1872. See also the Monthly Notices of the Royal Astronomical Society for May 1872.) An edition of Vlacq's work, containing many corrections, was issued at Leipzig in 1794 under the title Thesaurus Logarithmorum Completus by Jurij Vega.

Callet's seven-place table (Paris, 1795), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of interpolation, which were greatest in the early part of the table; and this addition was generally included in seven-place tables. The only important published extension of Vlacq's table was made by Mr. Sang 1871, whose table contained the seven-place logarithms of all numbers below 200,000.

Briggs and Vlacq also published original tables of the logarithms of the trigonometric functions.

Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Prony, by an original computation, under the auspices of the French republican government of the 1700s. This work, which contained the logarithms of all numbers up to 100,000 to nineteen places, and of the numbers between 100,000 and 200,000 to twenty-four places, exists only in manuscript, "in seventeen enormous folios," at the Observatory of Paris. It was begun in 1792; and "the whole of the calculations, which to secure greater accuracy were performed in duplicate, and the two manuscripts subsequently collated with care, were completed in the short space of two years." (English Cyclopaedia, Biography, Vol. IV., article "Prony.") Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.

To the modern student who has the benefit of a calculator, the work put into the tables just mentioned is a small indication of the importance of logarithms.

## Trivia

log is written so frequently in mathematics that it has its own Unicode glyph: ㏒ (code 13266).

## Source

Much of the history of logarithms is derived from The Elements of Logarithms with an Explanation of the Three and Four Place Tables of Logarithmic and Trigonometric Functions, by James Mills Peirce, University Professor of Mathematics in Harvard University, 1873.

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