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In mathematics, a logarithm (to Base ''b'') of a number ''x'' is written \log_b(x) or, if unambiguous, \log(x), and equals the Exponent ''y'' that satisfies ''x'' = ''b''''y''.

In other words

: y=\log_b(x)\,\!

is equivalent to

: x= b^y\,\!

The base ''b'' must be neither 0 nor 1 (nor a Root of 1 in the case of the extension to Complex Numbers , the Complex Logarithm ), and is typically 10, ''e'' , or 2. When ''x'' and ''b'' are further restricted to positive Real Number s, the logarithm is a unique real number.

For example, since

:3^4 = 3 imes 3 imes 3 imes 3 = 81, \,

: \log_3(81) = 4, \,

or, in words, the base-3 logarithm of 81 is 4, or the log base-3 of 81 is 4.


THE LOGARITHM AS A FUNCTION


The function log''b''(''x'') depends on both ''b'' and ''x'', but the term logarithm function (or '''logarithmic function''') in standard usage refers to a function of the form log''b''(''x'') in which the ''' Base ''' ''b'' is fixed and so the only argument is ''x''. Thus there is one logarithm function for each value of the base ''b'' (which must be positive and must differ from 1).
Viewed in this way, the base-b logarithm function is the Inverse Function of the Exponential Function ''b''''x''. The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.


INTEGER AND NON-INTEGER EXPONENTS


If ''n'' is a Positive Integer , ''b''''n'' signifies the Product of ''n'' factors equal to ''b'':

:\underbrace{b imes b imes \cdots imes b}_n.

However, if ''b'' is a positive real number not equal to 1, this definition can be extended to any Real Number ''n'' in a Field (see Exponentiation ). Similarly, the logarithm function can be defined for any positive real number. For each positive base ''b'' not equal to 1, there is one logarithm Function and one exponential function, which are inverses of each other.

Logarithms can reduce multiplication operations to addition, division to subtraction, exponentiation to multiplication, and roots to division. Therefore, logarithms are useful for making lengthy numerical operations easier to perform and, before the advent of Electronic Computer s, they were widely used for this purpose in fields such as Astronomy , Engineering , Navigation , and Cartography . They have important mathematical properties and are still widely used today.


BASES


The most widely used bases for logarithms are 10, the mathematical constant '' E '' ≈ 2.71828... and 2. When "log" is written without a base (''b'' missing from log''b''), the intent can usually be determined from context:


To avoid confusion, it is best to specify the base if there is any chance of misinterpretation.


Other notations


The notation "ln(''x'')" invariably means loge(''x''), i.e., the natural logarithm of ''x'', but the implied base for "log(''x'')" varies by discipline:

  • Mathematicians generally understand both "ln(''x'')" and "log(''x'')" to mean loge(''x'') and write "log10(''x'')" when the base-10 logarithm of ''x'' is intended. Calculus textbooks will occasionally write "lg(''x'')" to represent "log10(''x'')".


  • Many engineers, biologists, astronomers, and some others write only "ln(''x'')" or "loge(''x'')" when they mean the natural logarithm of ''x'', and take "log(''x'')" to mean log10(''x'') or, sometimes in the context of Computing , Log2 (''x'').


  • On most calculators, the LOG button is log10(''x'') and LN is loge(''x'').


  • In most commonly used computer Programming Language s, including C , C++ , Java , Fortran , Ruby , and BASIC , the "log" function returns the natural logarithm. The base-10 function, if it is available, is generally "log10."


  • Some people use Log(''x'') (capital ''L'') to mean log10(''x''), and 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.



This chaos, historically, originates from the fact that the natural logarithm has nice mathematical properties (such as its derivative being 1/''x'', and having a simple definition), while the base 10 logarithms, or decimal logarithms, were more convenient for speeding calculations (back when they were used for that purpose). Thus natural logarithms were only extensively used in fields like calculus while decimal logarithms were widely used elsewhere.

As recently as 1984, Paul Halmos in his "automathography" ''I Want to Be a Mathematician'' 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 , many mathematicians have adopted the "ln" notation, but most use "log".

In computer science, the base 2 logarithm is sometimes written as lg(''x''), as suggested by Edward Reingold and popularized by Donald Knuth . However, lg(''x'') is also sometimes used for the common log, and lb(''x'') for the binary log.2 In Russian literature, the notation lg(''x'') is also generally used for the base 10 logarithm.
In German, lg(''x'') also denotes the base 10 logarithm, while sometimes ld(''x'') or lb(''x'') is used for the base 2 logarithm.

The clear advice of the United States Department Of Commerce National Institute Of Standards And Technology is to follow the ISO standard ''Mathematical signs and symbols for use in physical sciences and technology, ISO 31-11:1992'', which suggests these notations:4
  • The notation "ln(''x'')" means loge(''x'');

  • The notation "lg(''x'')" means log10(''x'');

  • The notation "lb(''x'')" means log2(''x'').



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 log''e'' and log10). To find a logarithm with base ''b'', using any other base ''k'':

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

Moreover, this result implies that all logarithm functions (whatever the base) are Similar to each other. So to calculate the log with base 2 of the number 16 with a calculator:

: \log_2(16) = rac{\log(16)}{\log(2)}.


USES OF LOGARITHMS


Logarithms are useful in solving equations in which exponents are unknown. They have simple Derivative s, so they are often used in the solution of Integral s. The logarithm is one of three closely related functions. In the equation ''b''''n'' = ''x'', ''b'' can be determined with Radical s, ''n'' with logarithms, and ''x'' with Exponentials . See Logarithmic Identities for several rules governing the logarithm functions.


Science and engineering


Various quantities in science are expressed as logarithms of other quantities; see Logarithmic Scale for an explanation and a more complete list.

  • In Chemistry , the negative of the base-10 logarithm of the Concentration of Hydronium ions (H3O+, the form H+ takes in water) is the measure known as PH . The concentration of hydronium ions in neutral Water is 10−7 mol/L at 25 °C, hence a pH of 7.


  • The ''bel'' (symbol B) is a Unit of measure which is the base-10 logarithm of Ratio s, such as Power levels and Voltage levels. It is mostly used in Telecommunication , Electronics , and Acoustics . It is used, in part, because the ear responds logarithmically to acoustic power. The Bel is named after telecommunications pioneer Alexander Graham Bell . The '' Decibel '' (dB), equal to 0.1 bel, is more commonly used. The '' Neper '' is a similar unit which uses the natural logarithm of a ratio.







  • Many types of engineering and scientific data are typically graphed on Log-log or Semilog axes, in order to most clearly show the form of the data.



Exponential functions


The Exponential Function e^x, also written as \mathrm{exp}(x), is defined as the inverse of the natural logarithm. It is positive for every real argument x.

The operation of "raising b to a power p" for positive arguments b and all real exponents p is defined by

:b^p = \exp({p\ln b }).\,

The antilogarithm function is another name for the inverse of the logarithmic function. It is written antilog''b''(''n'') and means the same as b^n.


Easier computations


Logarithms switch the focus from normal numbers to exponents. As long as the same base is used, this makes certain operations easier; in this table, upper-case variables represent logs of corresponding lower-case variables:
These relations typically made such operations on two numbers much faster and the proper use of logarithms was an essential skill before multiplying Calculator s became available.

The \log(a b) = \log(a) +\log(b) equation is fundamental (it implies effectively the other three relations in a field) because it
describes an Isomorphism between the additive group and the '''multiplicative group''' of the field.

To multiply two numbers, one found the logarithms of both numbers on a table of Common Logarithm s, added them, and then looked up the result in the table to find the product. This is faster than multiplying them by hand, provided that more than two decimal figures are needed in the result. The table needed to get an accuracy of seven decimals could be fit in a big book, and the table for nine decimals occupied a few shelves.

The discovery of logarithms just before Newton's era had an impact in the scientific world which can be compared with the invention of the computer in the 20th century, because many calculations which were too laborious became feasible.

When the Chronometer was invented in the 18th century, logarithms allowed all calculations needed for astronomical navigation to be reduced to just additions, speeding the process by one or two orders of magnitude. A table of logarithms with five decimals, plus logarithms of trigonometric functions, was enough for most astronomical navigation calculations, and those tables fit in a small book.

To compute powers or roots of a number, the common logarithm of that number was looked up and multiplied or divided by the radix. .


Calculus


The Derivative of the natural logarithm function is

: rac{d}{dx} \ln(x) = rac{1}{x}.

By applying the change-of-base rule, the derivative for other bases is

: rac{d}{dx} \log_b(x) = rac{d}{dx} rac {\ln(x)}{\ln(b)} = rac{1}{x \ln(b)} = rac{\log_b(e)}{x}.

The Antiderivative of the natural logarithm ln(''x'') is

: \int \ln(x) \,dx = x \ln(x) - x + C,

and so the Antiderivative of the logarithm for other bases is

: \int \log_b(x) \,dx = x \log_b(x) - rac{x}{\ln(b)} + C = x \log_b \left( rac{x}{e} ight) + C.

''See also:'' , List Of Integrals Of Logarithmic Functions .


SERIES FOR CALCULATING THE NATURAL LOGARITHM


There are several series for calculating natural logarithms.''Handbook of Mathematical Functions'', National Bureau of Standards (Applied Mathematics Series no.55), June 1964, page 68. The simplest, though inefficient, is: