In mathematics, the logarithm of a number is the exponent by which

another fixed value, the base, must be raised to produce that number. For
example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the
3rd power: 1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is
the logarithm of x to base b, written logb x, so log10 1000 = 3. As a
single-variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.

Addition, multiplication, and exponentiation are three of the most

fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

by which tables of logarithms allow multiplication and division to be

reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

 Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}
Typical scientific calculators calculate the logarithms to bases 10 and
e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:

log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle

\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}
provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.

Addition, multiplication, and exponentiation are three of the most

fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula
log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log
_{b}(xy)=\log _{b}x+\log _{b}y,}

by which tables of logarithms allow multiplication and division to be

reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:

log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle

\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.

Addition, multiplication, and exponentiation are three of the most

fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

by which tables of logarithms allow multiplication and division to be

reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
 log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:
log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle
\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.

Addition, multiplication, and exponentiation are three of the most

fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}
by which tables of logarithms allow multiplication and division to be
reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:

log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle

\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.
Addition, multiplication, and exponentiation are three of the most
fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

by which tables of logarithms allow multiplication and division to be

reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
 For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:
log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle
\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.

Addition, multiplication, and exponentiation are three of the most

fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}
by which tables of logarithms allow multiplication and division to be
reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:

log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle

\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.
Addition, multiplication, and exponentiation are three of the most
fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

by which tables of logarithms allow multiplication and division to be

reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
 For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:
log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle
\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.

Addition, multiplication, and exponentiation are three of the most

fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}
by which tables of logarithms allow multiplication and division to be
reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:

log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle

\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.
Addition, multiplication, and exponentiation are three of the most
fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

by which tables of logarithms allow multiplication and division to be

reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
 For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:
log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle
\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.

Addition, multiplication, and exponentiation are three of the most

fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}
by which tables of logarithms allow multiplication and division to be
reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:

log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle

\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.
Addition, multiplication, and exponentiation are three of the most
fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

by which tables of logarithms allow multiplication and division to be

reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
 For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:
log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle
\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.

Addition, multiplication, and exponentiation are three of the most

fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}
by which tables of logarithms allow multiplication and division to be
reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:

log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle

\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.
Addition, multiplication, and exponentiation are three of the most
fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

by which tables of logarithms allow multiplication and division to be

reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
 For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:
log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle
\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.

Addition, multiplication, and exponentiation are three of the most

fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}
by which tables of logarithms allow multiplication and division to be
reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:

log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle

\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6] In mathematics,
the logarithm of a number is the exponent by which another fixed value,
the base, must be raised to produce that number. For example, the
logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power:
1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the
logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-
variable function, the logarithm to base b is the inverse of
exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is

commonly used in science and engineering. The natural logarithm has the
number e ≈ 2.718 as its base; its use is widespread in mathematics and
physics because of its very simple derivative. The binary logarithm uses
base 2 and is widely used in computer science, information theory, music
theory, and photography. When the base is unambiguous from the context or
irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of

simplifying calculations.[1] They were rapidly adopted by navigators,
scientists, engineers, surveyors, and others to perform high-accuracy
computations more easily. Using logarithm tables, tedious multi-digit
multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because the logarithm of a product is the sum
of the logarithms of the factors:

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

provided that b, x and y are all positive and b ≠ 1. The slide rule, also
based on logarithms, allows quick calculations without tables, but at
lower precision. The present-day notion of logarithms comes from Leonhard
Euler, who connected them to the exponential function in the 18th
century, and who also introduced the letter e as the base of natural
logarithms.[2]

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For

example, the decibel (dB) is a unit used to express ratio as logarithms,
mostly for signal power and amplitude (of which sound pressure is a
common example). In chemistry, pH is a logarithmic measure for the
acidity of an aqueous solution. Logarithms are commonplace in scientific
formulae, and in measurements of the complexity of algorithms and of
geometric objects called fractals. They help to describe frequency ratios
of musical intervals, appear in formulas counting prime numbers or
approximating factorials, inform some models in psychophysics, and can
aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to

other mathematical structures as well. However, in general settings, the
logarithm tends to be a multi-valued function. For example, the complex
logarithm is the multi-valued inverse of the complex exponential
function. Similarly, the discrete logarithm is the multi-valued inverse
of the exponential function in finite groups; it has uses in public-key
cryptography.
Motivation
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and
approaching minus infinity along the y-axis.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes
through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) =
3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does
not meet it.
Addition, multiplication, and exponentiation are three of the most
fundamental arithmetic operations. The inverse of addition is
subtraction, and the inverse of multiplication is division. Similarly, a
logarithm is the inverse operation of exponentiation. Exponentiation is
when a number b, the base, is raised to a certain power y, the exponent,
to give a value x; this is denoted

b y = x . {\displaystyle b^{y}=x.}

For example, raising 2 to the power of 3 gives 8: 2 3 = 8. {\displaystyle

2^{3}=8.}

The logarithm of base b is the inverse operation, that provides the

output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log
_{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a
positive real number. (If b is not a positive real number, both
exponentiation and logarithm can be defined but may take several values,
which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the

formula

log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log

_{b}(xy)=\log _{b}x+\log _{b}y,}

by which tables of logarithms allow multiplication and division to be

reduced to addition and subtraction, a great aid to calculations before
the invention of computers.
Definition

Given a positive real number b such that b ≠ 1, the logarithm of a

positive real number x with respect to base b[nb 1] is the exponent by
which b must be raised to yield x. In other words, the logarithm of x to
base b is the unique real number y such that b y = x {\displaystyle
b^{y}=x}.[3]

The logarithm is denoted "logb x" (pronounced as "the logarithm of x to

base b", "the base-b logarithm of x", or most commonly "the log, base b,
of x").

An equivalent and more succinct definition is that the function logb is

the inverse function to the function x ↦ b x {\displaystyle x\mapsto
b^{x}}.
Examples

log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.

Logarithms can also be negative: log 2 1 2 = − 1 {\textstyle \log
_{2}\!{\frac {1}{2}}=-1} since 2 − 1 = 1 2 1 = 1 2 . {\textstyle 2^{-
1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as
150 lies between 102 = 100 and 103 = 1000.
 For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1,
respectively.

Logarithmic identities
Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or

logarithmic laws, relate logarithms to one another.[4]
Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers

being multiplied; the logarithm of the ratio of two numbers is the
difference of the logarithms. The logarithm of the p-th power of a number
is p times the logarithm of the number itself; the logarithm of a p-th
root is the logarithm of the number divided by p. The following table
lists these identities with examples. Each of the identities can be
derived after substitution of the logarithm definitions x = b log b ⁡ x
{\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle
y=b^{\,\log _{b}y}} in the left hand sides. In the following formulas, ⁠ x
{\displaystyle x}⁠ and ⁠ y {\displaystyle y}⁠ are positive real numbers and ⁠
p {\displaystyle p}⁠ is an integer greater than 1.
Product, quotient, power, and root identities of logarithms Identity
Formula Example
Product log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log
_{b}(xy)=\log _{b}x+\log _{b}y} log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log
3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot
27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log
_{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} log 2 ⁡ 16 = log 2 64 4 =
log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log
_{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log
_{b}\left(x^{p}\right)=p\log _{b}x} log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6
log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log
_{2}2=6}
Root log b ⁡ x p = log b ⁡ x p {\textstyle \log
_{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} log 10 ⁡ 1000 = 1 2 log 10
⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac
{1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
Change of base

The logarithm logb x can be computed from the logarithms of x and b with
respect to an arbitrary base k using the following formula:[nb 2]

log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log

_{k}x}{\log _{k}b}}.}

Typical scientific calculators calculate the logarithms to bases 10 and

e.[5] Logarithms with respect to any base b can be determined using
either of these two logarithms by the previous formula:
log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle
\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log
_{e}b}}.}

Given a number x and its logarithm y = logb x to an unknown base b, the

base is given by:

b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},}

which can be seen from taking the defining equation x = b log b ⁡ x = b y

{\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y .
{\displaystyle {\tfrac {1}{y}}.}
Particular bases
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠, 2, and e

Among all choices for the base, three are particularly common. These are
b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and
b = 2 (the binary logarithm). In mathematical analysis, the logarithm
base e is widespread because of analytical properties explained below. On
the other hand, base 10 logarithms (the common logarithm) are easy to use
for manual calculations in the decimal number system:[6]