Logarithm From Wikipedia, the free encyclopedia Logarithm functions, graphed for various bases: red is to base e, green is to base 10, and purple is to base 1.7. Each tick on the axes is one unit. Logarithms of all bases pass through the point (1, 0), because any non-zero number raised to the power 0 is 1, and through the points (b, 1) for base b, because a number raised to the power 1 is itself. The curves approach the y-axis but do not reach it because of the singularity at x = 0 (a vertical asymptote ). The 1797 Encyclopædia Britannica explains logarithms as "a series of numbers in arithmetical progression, corresponding to others in geometrical progression; by means of which, arithmetical calculations can be made with much more ease and expedition than otherwise." In mathematics , the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 3 is the power to which ten must be raised to produce 1000: 10 3 = 1000, so log101000 = 3. Only positive real numbers have real number logarithms; negative and complex numbers have complex logarithms . The logarithm of x to the base b is written logb(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y, The bases used most often are 10 for the common logarithm , e for the natural logarithm , and 2 for the binary logarithm .
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LogarithmFrom Wikipedia, the free encyclopedia
Logarithm functions, graphed for various bases: red is to base e, green is to base 10, and purple is to base 1.7. Each tick on the axes is one unit. Logarithms
of all bases pass through the point (1, 0), because any non-zero number raised to the power 0 is 1, and through the points (b, 1) for base b, because a
number raised to the power 1 is itself. The curves approach the y-axis but do not reach it because of the singularity at x = 0 (a vertical asymptote).
The 1797 Encyclopædia Britannica explains logarithms as "a series of numbers in arithmetical progression, corresponding to others in geometrical
progression; by means of which, arithmetical calculations can be made with much more ease and expedition than otherwise."
In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce
that number. For example, the logarithm of 1000 to base 10 is 3, because 3 is the power to which ten must be raised to produce 1000:
103 = 1000, so log101000 = 3. Only positive real numbers have real number logarithms; negative and complex numbers have complex
logarithms.
The logarithm of x to the base b is written logb(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y,
The bases used most often are 10 for the common logarithm, e for the natural logarithm, and 2 for the binary logarithm.
An important feature of logarithms is that they reduce multiplication to addition, by the formula:
That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers.
Similarly, logarithms reduce division to subtraction by the formula:
That is, the logarithm of the quotient of two numbers is the difference between the logarithms of those numbers.
The use of logarithms to facilitate complicated calculations was a significant motivation in their original development. Logarithms have applications in fields as diverse as statistics, chemistry, physics, astronomy, computer science, economics, music, and engineering.
ogarithm of positive real numbers
[Definition
The graph of the function f(x) = 2x (red) together with a depiction of log2(3) ≈ 1.58.
The logarithm of a positive real number y with respect to another positive real number b, where b is not equal to 1, is the real
number x such that
that is, the x-th power of b must equal y.[1][2]
The logarithm x is denoted logb(y). (Some European countries write blog(y) instead.[3]) The number b is referred to as the base.
For b = 2, for example, this means
since 23 = 2 · 2 · 2 = 8. The logarithm may be negative, for example
how many digits needed to write that number in that base. For instance, the common logarithm of a number x tells how
many numerical digits x has: when
n − 1 ≤ log10(x) < n,
with an integer n then x has n decimal digits. Since we write numbers in base 10, mental math is thus easier with
the common log[citation needed], making it attractive to many engineers. The approximation 210 ≈ 103 leads to the approximations
3 dB per octave (power doubling) – a useful result that occurs with the use of log10.
The natural logarithm, loge(x) is the one with base b = e. It has many "natural" properties related to
its analytical behavior explained below. It is found in mathematical analysis, statistics, economics and some
engineering fields. For example, Euler's identity is important to fields that deal with cyclic components. The natural
logarithm of x is often written "ln(x)", instead of loge(x) especially in disciplines where it isn't written "log(x)". However, some
mathematicians disapprove of this notation. In his 1985 autobiography, Paul Halmos criticized what he considered the
"childish ln notation," which he said no mathematician had ever used.[4] In fact, the notation was invented by a
mathematician, Irving Stringham, professor of mathematics at University of California, Berkeley, in
1893.[5][6]
The binary logarithm with base b = 2 is used computer science and information theory. Computers
ubiquitously use binary storage with bits as the basic unit and it takes at least ⌊log2(n)⌋+1 bits to store the integer n. Likewise,
a binary search through a sorted list of size n takes ⌊log2(n)⌋+1 steps. Properties like this come up repeatedly in these
domains.
[edit]Implicit bases
Instead of writing logb(x), it is common to omit the base, log(x), when intended base can be determined from context. In
mathematics and many programming languages,[7] "log(x)" is usually understood to be the natural logarithm.
Engineers, biologists and astronomers often define "log(x)" to be the common logarithm, log10(x), while computer scientists
often choose "log(x)" to be the binary logarithm, log2(x).
On most calculators, the "log" button is log10(x) and "ln" is loge(x). The International Organization for Standardization (ISO 31-11) suggests the notations "ln(x)", "lg(x)", "lb(x)" for loge(x), log10(x), and log2(x),
respectively.[8]
The base, b, used by the supplied logarithm function can be explicitly determined using the following identity (subject to the
inherent computational accuracy errors).
This follows from the change-of-base formula above.
[edit]Computer science
In computer science, the base-2 logarithm is sometimes written as lg(x), as suggested by Edward Reingold and
popularized by Donald Knuth. However, lg(x) is also sometimes used for the common logarithm, and lb(x) for the base-2
logarithm.[9] In Russian literature, the notation lg(x) is also generally used for the base-10 logarithm.[10] In German, lg(x) also
The principal branch of the complex logarithm, Log(z). The hue of the color shows the argument of Log(z), the saturation (intensity) of the color shows the
absolute value of the complex logarithm.
The above definition of logarithms of positive real numbers can be extended to complex numbers. This generalization known
as complex logarithm requires more care than the logarithm of positive real numbers. Any complex number z can be
represented as z = x + iy, where x and y are real numbers and i is the imaginary unit. The intent of the logarithm is—as with the
natural logarithm of real numbers above— to find a complex number a such that the exponential of a equals z:
ea = z.
Polar form of complex numbers
This can be solved for any z ≠ 0, however there are multiple solutions. To see this, it is convenient to use the polar
form of z, i.e., to express z as
z = r(cos(φ) + i sin(φ))
where is the absolute value of z and φ = arg(z) an argument of z, that is, is any angle such
thatx = r cos(φ) and y = r sin(φ). Geometrically, the absolute value is the distance of z to the origin and the argument is the
angle between the x-axis and the line passing through the origin and z. The argument φ is not unique: φ' = φ + 2π is an
argument, too, since "winding" around the circle counter-clock-wise once corresponds to adding 2π (360 degrees) to φ.
Logarithms appear in Benford's law, a empirical description of the occurrence of digits in certain real-life data sources, such as
heights of buildings. The probability that the first decimal digit of the data in question is d (from 1 to 9) equals
P(d) = log10(d + 1) − log10(d),
irrespective of the unit of measurement. This can be used to detect fraud in accounting.[14]
[edit]Music
Logarithms appear in measuring musical intervals: the interval between two notes in semitones is the base-21/12 logarithm of
the frequency ratio. For finer encoding, for example for non-equal temperaments, intervals are also expressed
in cents (hundredths of an equally-tempered semitone). The interval between two notes in cents is the base-21/1200 logarithm of the frequency ratio (or 1200 times the base-2 logarithm). The table below lists some musical intervals together with the frequency ratios and their logarithms.
Interval (two tones are played at the
same time)
1/72 tone play (help·info)
Semitone play
Just major third
play
Major third play Tritone play
Octave
play
Frequency ratio r 2
, i.e., corresponding number of semitones
1/6 1 ≈ 3.86 4 6 12
, i.e., corresponding number of cents
16.67 100≈
386.31
400 6001200
[edit]Related operations and generalizations
The cologarithm of a number is the logarithm of the reciprocal of the number: cologb(x) = logb(1/x) = −logb(x). This terminology is
found primarily in older books.[15]
The antilogarithm function antilogb(y) is the inverse function of the logarithm function logb(x); it can be written in closed form
as by. The antilog notation was common before the advent of modern calculators and computers: tables of antilogarithms to the
base 10 were useful in carrying out computations by hand.[16] Today's applications of antilogarithms include certain electronic
circuit components known asantilog amplifiers,[17] or the calculation of equilibrium constants of reactions involving electrode