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In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number.

For example, the logarithm of 1000 to the base 10 is 3, because 10 raised to the power of 3 is 1000; the base 2 logarithm of 32 is 5 because 2 to the power 5 is 32.

The logarithm of x to the base b is written log_b(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y,

${\text{ if }}b^{y}=x,{\text{ then }}y=\log _{b}(x)\,.$

An important feature of logarithms is that they reduce multiplication to addition, by the formula:

$\log(xy)=\log x+\log y\,.$

That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers. The use of logarithms to facilitate complex calculations was a significant motivation in their original development.

Properties of the logarithm

Template:Main article When x and b are restricted to positive real numbers, log_b(x) is a unique real number. The magnitude of the base b must be neither 0 nor 1; the base used is typically 10, e, or 2. Logarithms are defined for real numbers and for complex numbers.

The major property of logarithms is that they map multiplication to addition. This ability stems from the following identity:

$b^{x}\times b^{y}=b^{x+y}\ ,$

which by taking logarithms becomes

$\log _{b}\left(b^{x}\times b^{y}\right)=\log _{b}\left(b^{x+y}\right)$ $\ =x+y=\log _{b}\left(b^{x}\right)+\log _{b}\left(b^{y}\right).\$

For example,

$4=2^{2}\,\Rightarrow \,\log _{2}(4)=2\,,$

$8=2^{3}\,\Rightarrow \,\log _{2}(8)=3\,,$

$\log _{2}(32)=\log _{2}(4\times 8)=\log _{2}(4)+\log _{2}(8)=2+3=5\,.$

A related property is reduction of exponentiation to multiplication. Using the identity:

$c=b^{\log _{b}(c)}\ ,$

it follows that c to the power p (exponentiation) is:

$c^{p}=\left(b^{\log _{b}(c)}\right)^{p}=b^{p\log _{b}(c)}\ ,$

or, taking logarithms:

$\log _{b}\left(c^{p}\right)=p\log _{b}(c)\ .$

In words, to raise a number to a power p, find the logarithm of the number and multiply it by p. The exponentiated value is then the inverse logarithm of this product; that is, number to power = b^product.

For example,

$\log _{2}(64)=\log _{2}(4^{3})=3\log _{2}(4)=6\,.$

Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division. For example,

$\log _{2}(16)=\log _{2}\left({\frac {64}{4}}\right)=\log _{2}(64)-\log _{2}(4)=6-2=4\,,$

$\log _{2}({\sqrt[{3}]{4}})={\frac {1}{3}}\log _{2}(4)={\frac {2}{3}}\,.$

Logarithms make lengthy numerical operations easier to perform. The whole process is made easy by using tables of logarithms, or a slide rule, antiquated now that calculators are available. Although the above practical advantages are not important for numerical work today, they are used in graphical analysis (see Bode plot).

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:

• natural logarithm (log_e, ln, log, or Ln) in mathematical analysis, statistics, economics and some engineering fields. The reasons to consider e the natural base for logarithms, though perhaps not obvious, are numerous and compelling.
• common logarithm (log₁₀ or simply log; sometimes lg) in various engineering fields, especially for power levels and power ratios, such as acoustical sound pressure, and in logarithm tables to be used to simplify hand calculations
• binary logarithm (log₂; sometimes lg, lb, or ld), in computer science and information theory
• indefinite logarithm (Log or [log ] or simply log) when the base is irrelevant, e.g. in complexity theory when describing the asymptotic behavior of algorithms in big O notation.

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 log_e(x), i.e., the natural logarithm of x, but the implied base for "log(x)" varies by discipline:

• Mathematicians understand "log(x)" to mean log_e(x). Calculus textbooks will occasionally write "lg(x)" to represent "log₁₀(x)".

• Many engineers, biologists, astronomers, and some others write only "ln(x)" or "log_e(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log₁₀(x) or, in computer science, log₂(x).

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

• In most commonly used computer programming languages, 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.

• In some European countries, a frequently used notation is ^blog(x) instead of log_b(x).

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.

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. 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:

• The notation "ln(x)" means log_e(x);
• The notation "lg(x)" means log₁₀(x);
• The notation "lb(x)" means log₂(x).