Although logarithms are now understood as exponents, Napier himself did not think of them in this way.
He developed his theory of logarithms geometrically,
thinking of them as a ratio of distances traveled by two
moving objects, one moving along a straight line at a
speed of 1 unit per second, and the other moving along
a line segment 1 unit long and with speed changing
according to its distance from the endpoint it is
approaching. Napier chose the name logarithm from
the Greek words logos for “ratio” and arithmos for
“number.” It was not until the end of the 17th century
that mathematicians recognized that logarithms were,
in fact, exponents.
After Napier published his work in 1614, English
mathematician HENRY BRIGGS (1561–1630) suggested
to Napier that, like our number system, logarithms
should be based on the number 10. Napier agreed that
this would indeed simplify matters, and b = 10 was
then deemed the preferred base for logarithms. Base-10
logarithms are today called common logarithms or
Briggs’s logarithms. The common logarithm of N is
simply denoted log N or lg N.
In 1624 Briggs published tables listing values of
common logarithms for the numbers 1 to 20,000 and
90,000 to 100,000, inclusive. The values for the numbers 20,000–90,000 were completed after Briggs’s
death by Dutch mathematician Adriaan Vlacq.
Swiss watchmaker Jobst Bürgi, maker of astronomical instruments, also conceived of logarithms to facilitate the multiplication of large numbers. However,
since Napier published his work first, the credit for
their discovery was not given to Bürgi.
A number with a given value for its logarithm is
called the antilogarithm, or antilog, of that value. The
base-10 antilog of a value x is 10
x
.
The common logarithms of the numbers 3.7, 370,
370,000, for example, differ by whole numbers: log 3.7
≈ 0.5682, log 370 = log(3.7 × 100) ≈ 0.5682 + 2 =
2.5682, and log 370,000 = log(3.7 × 10
5
) ≈ 5.5682.
The decimal part of a logarithmic value is called the
mantissa, and the integer part is called the characteristic of the logarithm. (For example, the three logarithms
above each have mantissa 0.5682 and characteristics 0,
2, and 5, respectively.) Logarithmic tables from the past
listed only the mantissas of numbers from 1 to 10. The
logarithm of any other number can then be computed
by adding the appropriate integer to represent the
power of 10 needed.
In CALCULUS it is convenient to work with logarithms of base e. The number e is an irrational number
with value approximately 2.718281828… Logarithms
of base e are called natural logarithms, and a logarithm
of base e is denoted ln. Thus, ln(e
3 ) = 3, for example.
If one is willing to work with complex numbers,
then it is possible to give meaning to the logarithm of a
negative number. For example, EULER’S FORMULA tells
us that e
iπ = –1. Consequently ln(–1) = iπ. Going further, extending to logarithms of complex numbers, we
have, for instance,
. In this
setting, all nonzero numbers—real and complex—have
logarithms.
See also E; EXPONENTIAL FUNCTION; LOGARITHMIC
FUNCTION; LOGARITHMIC SCALE; SLIDE RULE.
logarithmic function Any CONTINUOUS FUNCTION
f(x), not identically zero, defined for positive values of
x with the property that f(a · x) = f(a) + f(x) for all positive values a and x, is called a logarithmic function.
Such functions are said to “convert multiplication into
addition.” The series of observations below shows that
every logarithmic function is given by a LOGARITHM:
f(x) = log b x for some positive base b.
1. All logarithmic functions f(x) satisfy f(1) = 0.
(This follows from the observation: f(1) = f(1.1) =
f(1) + f(1).)
2. All logarithmic functions satisfy
.
Consequently, the logarithmic functions give both
positive and negative outputs.
(This follows from the observation:
.)
3. For any logarithmic function there is a number b for
which f(b) = 1.
(We have f(1) = 0 and that it is possible to choose a value
a such that f(a) is positive. One of the values f(a), f(a
2
) =
f(a) + f(a), f(a
3
) = f(a) + f(a) + f(a), … will be greater than
one. Thus it is possible to find a number c with f(c) > 1.
By the INTERMEDIATE-VALUE THEOREM, there must be a
value b between 1 and c, so that f(b) = 1.)
1
+
f a f a
( ) ( )
0
1
1
=
=
⋅ =
f
f a a
( ) (
)
f a
f a
( )
( )
1 = −
ln ln
ln( )
i
i
=
−
( ) = − =
1
1
2
1
2
π
320 logarithmic function
He developed his theory of logarithms geometrically,
thinking of them as a ratio of distances traveled by two
moving objects, one moving along a straight line at a
speed of 1 unit per second, and the other moving along
a line segment 1 unit long and with speed changing
according to its distance from the endpoint it is
approaching. Napier chose the name logarithm from
the Greek words logos for “ratio” and arithmos for
“number.” It was not until the end of the 17th century
that mathematicians recognized that logarithms were,
in fact, exponents.
After Napier published his work in 1614, English
mathematician HENRY BRIGGS (1561–1630) suggested
to Napier that, like our number system, logarithms
should be based on the number 10. Napier agreed that
this would indeed simplify matters, and b = 10 was
then deemed the preferred base for logarithms. Base-10
logarithms are today called common logarithms or
Briggs’s logarithms. The common logarithm of N is
simply denoted log N or lg N.
In 1624 Briggs published tables listing values of
common logarithms for the numbers 1 to 20,000 and
90,000 to 100,000, inclusive. The values for the numbers 20,000–90,000 were completed after Briggs’s
death by Dutch mathematician Adriaan Vlacq.
Swiss watchmaker Jobst Bürgi, maker of astronomical instruments, also conceived of logarithms to facilitate the multiplication of large numbers. However,
since Napier published his work first, the credit for
their discovery was not given to Bürgi.
A number with a given value for its logarithm is
called the antilogarithm, or antilog, of that value. The
base-10 antilog of a value x is 10
x
.
The common logarithms of the numbers 3.7, 370,
370,000, for example, differ by whole numbers: log 3.7
≈ 0.5682, log 370 = log(3.7 × 100) ≈ 0.5682 + 2 =
2.5682, and log 370,000 = log(3.7 × 10
5
) ≈ 5.5682.
The decimal part of a logarithmic value is called the
mantissa, and the integer part is called the characteristic of the logarithm. (For example, the three logarithms
above each have mantissa 0.5682 and characteristics 0,
2, and 5, respectively.) Logarithmic tables from the past
listed only the mantissas of numbers from 1 to 10. The
logarithm of any other number can then be computed
by adding the appropriate integer to represent the
power of 10 needed.
In CALCULUS it is convenient to work with logarithms of base e. The number e is an irrational number
with value approximately 2.718281828… Logarithms
of base e are called natural logarithms, and a logarithm
of base e is denoted ln. Thus, ln(e
3 ) = 3, for example.
If one is willing to work with complex numbers,
then it is possible to give meaning to the logarithm of a
negative number. For example, EULER’S FORMULA tells
us that e
iπ = –1. Consequently ln(–1) = iπ. Going further, extending to logarithms of complex numbers, we
have, for instance,
. In this
setting, all nonzero numbers—real and complex—have
logarithms.
See also E; EXPONENTIAL FUNCTION; LOGARITHMIC
FUNCTION; LOGARITHMIC SCALE; SLIDE RULE.
logarithmic function Any CONTINUOUS FUNCTION
f(x), not identically zero, defined for positive values of
x with the property that f(a · x) = f(a) + f(x) for all positive values a and x, is called a logarithmic function.
Such functions are said to “convert multiplication into
addition.” The series of observations below shows that
every logarithmic function is given by a LOGARITHM:
f(x) = log b x for some positive base b.
1. All logarithmic functions f(x) satisfy f(1) = 0.
(This follows from the observation: f(1) = f(1.1) =
f(1) + f(1).)
2. All logarithmic functions satisfy
.
Consequently, the logarithmic functions give both
positive and negative outputs.
(This follows from the observation:
.)
3. For any logarithmic function there is a number b for
which f(b) = 1.
(We have f(1) = 0 and that it is possible to choose a value
a such that f(a) is positive. One of the values f(a), f(a
2
) =
f(a) + f(a), f(a
3
) = f(a) + f(a) + f(a), … will be greater than
one. Thus it is possible to find a number c with f(c) > 1.
By the INTERMEDIATE-VALUE THEOREM, there must be a
value b between 1 and c, so that f(b) = 1.)
1
+
f a f a
( ) ( )
0
1
1
=
=
⋅ =
f
f a a
( ) (
)
f a
f a
( )
( )
1 = −
ln ln
ln( )
i
i
=
−
( ) = − =
1
1
2
1
2
π
320 logarithmic function