The document explains logarithms, their functions, and applications in various fields such as mathematics and engineering. It describes how logarithms convert multiplication into addition and division into subtraction, facilitating easier calculations. The document also covers different bases for logarithms, including common, natural, and binary logarithms, and their properties related to graphs and identities.

1. 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:
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:

2. 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 b
log(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
since

3. The right image shows how to determine (approximately) the logarithm. Given the graph (in red) of the function f(x) = 2x
, the logarithm
log2(y) is the For any given number y (y = 3 in the image), the logarithm of y to the base 2 is the x-coordinate of the intersection point of
the graph and the horizontal line intersecting the vertical axis at 3.
Above, the logarithm has been defined to be the solution of an equation. For this to be meaningful, it is thus necessary to ensure that
there is always exactly one such solution. This is done using three properties of the function f(x) = bx
: in the case b > 1, this function f(x)
is strictly increasing, that is to say, f(x) increases when x does so. Secondly, the function takes arbitrarily big values and arbitrarily small
positive values. Thirdly, the function is continuous. Intuitively, the function does not "jump": the graph can be drawn without lifting the
pen. These properties, together with the intermediate value theorem ofelementary calculus ensure that there is indeed exactly one
solution x to the equation
f(x) = bx
= y,
for any given positive y. When 0 < b < 1, a similar argument is used, except that f(x) = bx
is decreasing in that case.
[edit]Identities
Main article: Logarithmic Identities
The above definition of the logarithm implies a number of properties.
[edit]Logarithm of products
Logarithms map multiplication to addition. That is to say, for any two positive real numbers x and y, and a given positive base b, the
identity
logb(x · y) = logb(x) + logb(y).
For example,
log3(9 · 27) = log3(243) = 5,
since 35
= 243. On the other hand, the sum of log3(9) = 2 and log3(27) = 3 also equals 5. In general, that identity is derived from the
relation of powers and multiplication:
bs
· bt
= bs + t.
Indeed, with the particular values s = logb(x) and t = logb(y), the preceding equality implies
logb(bs
· bt
) = logb(bs + t
) = s + t = logb(bs
) + logb(bt
).
By virtue of this identity, logarithms make lengthy numerical operations easier to perform by converting multiplications to additions. The
manual computation process is made easy by using tables of logarithms, or a slide rule. The property of common logarithms pertinent to the
use of log tables is that any decimal sequence of the same digits, but different decimal-point positions, will have identical mantissas and
differ only in their characteristics.
Logarithm of powers
A related property is reduction of exponentiation to multiplication. Another way of rephrasing the definition of the logarithm is to write
x = blog
b
(x)
.

4. Raising both sides of the equation to the p-th power (exponentiation) shows
xp
= (blog
b
(x)
)p
= bp · log
b
(x)
.
thus, by taking logarithms:
logb(xp
) = p logb(x).
In prose, the logarithm of the p-th power of x is p times the logarithm of x. As an example,
log2(64) = log2(43
) = 3 · log2(4) = 3 · 2 = 6.
Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction,
and roots to division. For example,
Change of base
The above rule for a logarithm of a power can be used to derive a relation between logarithms with respect to different bases:
In fact, the left hand side of the above is the unique number a such that ba
= x. Therefore
logk(x) = logk(ba
) = a · logk(b).
The general restriction b ≠ 1 implies logkb ≠ 0, since b0
= 1. Thus, dividing the preceding equation by logkb shows the above
formula.
As a practical consequence, logarithms with respect to any base k can be calculated, e.g. using a calculator, if logarithms
to the base b are available. From a more theoretical viewpoint, this result implies that the graphs of all logarithm functions
(whatever the base b) are similar to each other.
One way of viewing the change-of-base formula is to say that in the expression
the two bs "cancel", leaving logk x.
[edit]Bases
While the definition of logarithm applies to any positive real number b (1 is excluded, though), a few particular choices for b are
more commonly used. These are b = 10, b = e (the mathematical constant ≈ 2.71828…), and b = 2. The different standards
come about because of the different properties preferred in different fields.
For b = 10, the logarithm log10 is called common logarithm. It appears in various engineering fields, especially for
power levels and power ratios, such as acoustical sound pressure, and in logarithm tables to be used to simplify
hand calculations. Its use is historically grounded (see dB)[citation needed]
. The logarithm of a number in a given base can show

5. 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.

6. [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
denotes the base-10 logarithm, while sometimes ld(x) or lb(x) is used for the base-2 logarithm. The PL/I Programming language
uses log2(x) for the base-2 logarithm.
[edit]Equivalence of logarithms
The above identity relating logarithms with respect to different bases shows that the difference between logarithms to different
bases is one of scale. For example, the unit decibel (dB) refers to a common logarithm (b = 10). Alternatively, neper are
based on a natural logarithm. Another example from information theory: calculations carried out using log2 will lead to
results in bits, which has an intuitive meaning; corresponding calculations carried out using loge will lead to results
in nats which may lack this intuitive interpretation. However, the change amounts to a factor of loge2≈0.69—twice as many
values can be encoded with one additional bit, which corresponds to an increase of about 0.69 nats.
Logarithms to a base less than 1 have a negative scale, or a flip about the x axis, relative to logarithms of base greater than 1.
In disciplines where the scale is irrelevant, the term indefinite logarithm refers to point of view. An example
is complexity theory which describes the asymptotic behavior of algorithms in big O notation. It often
makes statements like "the behavior of the algorithm is logarithmic", but does not measure of performance of the algorithm in a
given situation.
Logarithm as a function
The graph of the logarithm function logb(x) (green) is obtained by reflecting the one of the function bx
(red) at the diagonal line (x = y).
The expression logb(x) depends on both b and x, but the term logarithm function (or logarithmic function) refers to a function of
the form logb(x) in which the base bis fixed and x is variable, thus yielding a function that assigns to any x its logarithm logb(x).
The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function. The
above definition of logarithms were done indirectly by means of the exponential function. A compact way of rephrasing that

7. definition is to say that the base-b logarithm function is the inverse function of the exponential function bx
: a point (t, u = b(t)
) on
the graph of the exponential function yields a point (u, t = logbu) on the graph of the logarithm and vice versa. Geometrically,
this corresponds to the statement that the points correspond one to another upon reflecting them at the diagonal line x = y.
Using this relation to the exponential function, calculus tells that the logarithm function is continuous (it does not "jump", i.e., the
logarithm of x changes only little when x varies only little). What is more, it is differentiable (intuitively, this means that the graph
of logb(x) has no sharp "corners").
[edit]Integral representation of the natural logarithm
The natural logarithm of t is the shaded area underneath the graph of the functionf(x) = 1/x (reciprocal of x).
When b = e (Euler's number), the natural logarithm ln(t) = loge(t) can be shown to satisfy the following identity:
In prose, the natural logarithm of t agrees with the integral of 1/x dx from 1 to t, that is to say, the area between the x-
axis and the function 1/x, ranging from x = 0 tox = t. This is depicted at the right. Some authors use the right hand side of
this equation as a definition of the natural logarithm.[citation needed]
The formulae of logarithms of products and powers
mentioned above can be derived directly from this presentation of the natural logarithm. The product formula ln(tu) = ln(t)
+ ln(u) is deduced in the following way:
The "=" labelled (1) used a splitting of the integral into two parts, the equality (2) is a change of variable (w = x/t).
This can also be understood geometrically. The integral is split into two parts (shown in yellow and blue). The key
point is this: rescaling the left hand blue area in vertical direction by the factor t and shrinking it by the same factor
in the horizontal direction does not change its size. Moving it appropriately, the area fits the graph of the function
1/x again. Therefore, the left hand area, which is the integral of f(x) from t to tu is the same as the integral
from 1 to u:

8. A visual proof of the product formula of the natural logarithm.
The power formula ln(tr
) = r ln(t) is derived similarly:
The second equality is using a change of variables, w := x1/r
, while the third equality follows from integration
by substitution.
[edit]Derivative and antiderivative
Further information: List of integrals of logarithmic functions
The fundamental theorem of calculus applied to the above integral formula shows that the derivative of the
natural logarithm function is
More generally, by the chain rule, the derivative with a generalised functional argument f(x) is
For this reason the quotient at right hand side is called logarithmic derivative of f.
The antiderivative of the natural logarithm ln(x) is
Calculation
[edit]Taylor series
There are several series for calculating natural logarithms.[11]
For all complex numbers z satisfying |1 − z| < 1, a simple, though
inefficient, series is
Actually, this is the Taylor series expansion of the natural logarithm at z = 1. It is derived from the geometric series (with |x| < 1)
Integration of both sides yields

9. and the above expression is obtained by substituting x by 1 − z (so that 1 − x = z).
[edit]More efficient series
A more efficient series is
for z with positive real part.
To derive this series, we begin by substituting −x for x and get
Subtracting, we get
Letting and thus , we get
The series converges most quickly if z is close to 1. For high-precision calculations, we can first obtain a low-accuracy
approximation y ≈ ln(z), then let A = z/exp(y), where exp(y) can be calculated using the exponential series, which converges
quickly provided y is not too large. Then ln(z) = y + ln(A), where A is close to 1 as desired. Larger z can be handled by
writing z = a × 10b
, whence ln(z) = ln(a) + b × ln(10) (using 10 as an example base). High precision calculations can be first
obtained by low accuracy as mentioned above, this helps in the mathematical process.
[edit]Computation using significands
In most computers, real numbers are modelled by floating points, which are usually stored as
x = m · 2n
.
In this representation m is called significand and n is the exponent. Therefore
logb(x) = logb(m) + n logb(2),
so that in order to compute the logarithm of x, it suffices to calculate logb(m) for some m such that 1 ≤ m < 2. Having m in this
range means that the value u = (m − 1)/(m + 1) is always in the range 0 ≤ u < 1/3. Some machines use the significand in the

10. range 0.5 ≤ m < 1 and in that case the value for u will be in the range −1/3 < u ≤ 0. In either case, the series is even easier to
compute.
The binary (base b = 2) logarithm of a number x can be approximated using the Binary logarithm Algorithm.
Logarithm of a negative or complex number
Main article: Complex logarithm
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

11. 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.
However, there is exactly one argument φ such that −π < φ and φ ≤ π, called the principal argument and denoted Arg(z) (with a
capital A).
It is a fact proven in complex analysis that z = reiφ
. Consequently,
a = ln(r) + i φ
is such that the a-th power of e equals z, so that a qualifies being called logarithm of z. When φ is chosen to be the principal
argument, a is called principal value of the logarithm, denoted Log(z). The principal argument of any positive real number is 0;
hence the principal logarithm of such a number is always real and equals the natural logarithm. The graph at the right depicts
Log(z). The discontinuity, i.e., jump in the hue at the negative part of the x-axis is due to jump of the principal argument at this
locus. This behavior can only be circumvented by dropping the range restriction on φ. Then, however, both the argument
of z and, consequently, its logarithm become multi-valued functions. In fact, any logarithm of z can be shown to be of the form
ln(r) + i (φ + 2nπ),
where φ is the principal argument Arg(z). Analogous formula for principal values of logarithm of products and powers for
complex numbers do in general not hold.
For any complex number b ≠ 0 and 1, the complex logarithm logb(z) with base b is defined as ln(z)/ln(b), the principal value of
which is given by the principal values of ln(z) and ln(b).
[edit]Uses and occurrences
A nautilus displaying a logarithmic spiral.
Logarithms appear in many places, within and outside mathematics. The logarithmic spiral, for example, appears
(approximately) in various guises in nature, such as the shells of nautilus.
[edit]Calculation of products, powers and roots
Logarithms can be used to reduce multiplications and exponentiations to additions. This fact was the historical motivation for
logarithms. The use of logarithms was an essential skill until computers and calculators became available. Indeed the discovery
of logarithms, just before Newton's era, had an impact in the scientific world that can be compared with that of the advent of
computers in the 20th century because it made feasible many calculations that had previously been too laborious.[citation needed]

12. The product of two numbers c and d can be calculated by the following formula:
c · d = blog
b
c
· blog
b
d
= b(log
b
c + log
b
d)
.
Using a table of logarithms, logbc and logbd can be looked up. After calculating their sum, an easy operation, the antilogarithm
of that sum is looked up in a table, which is the desired product. For manual calculations that demand any appreciable
precision, this process, requiring three lookups and a sum, is much faster than performing the multiplication. To achieve seven
decimal places of accuracy requires a table that fills a single large volume; a table for nine-decimal accuracy occupies a few
shelves. The precision of the approximation can be increased byinterpolating between table entries.
Divisions can be performed similarly: Moreover,
cd
= b(log
b
c) · d
reduces the exponetiation to looking up the logarithm of c, multiplying it with d (possibly using the previous method) and looking
up the antilogarithm of the product. Roots can be calculated this way, too, since .
One key application of these techniques was celestial navigation. Once the invention of the chronometer made possible the
accurate measurement of longitude at sea, mariners had everything necessary to reduce their navigational computations to
mere additions. A five-digit table of logarithms and a table of the logarithms of trigonometric functions sufficed for most
purposes, and those tables could fit in a small book. Another critical application with even broader impact was the slide rule,
until the 1970s an essential calculating tool for engineers and scientists. The slide rule allows much faster computation than
techniques based on tables, but provides much less precision (although slide rule operations can be chained to calculate
answers to any arbitrary precision).
Logarithmic scale
Main article: Logarithmic scale
Various quantities in science are expressed as logarithms of other quantities, a concept known as logarithmic scale. It applies
in various situations where a given quantity ranges from 1 to 10,000,000, say, in a way such that a change of the value from 1
to 2 is as important as from 100,000 to 200,000. Considering the logarithm instead of the quantity itself reduces such ranges to
smaller ones. Moreover, ratios between different values correspond to differences in their logarithms.
For example, in chemistry, the negative of the base-10 logarithm of the activity of hydronium ions (H3O+
, the form H+
takes in
water) is the measure known as pH. The activity of hydronium ions in neutralwater is 10−7
mol/L at 25 °C, hence a pH of
7. Vinegar, on the other hand, has a pH of about 3. The difference of 4 corresponds to a ratio of 104
of the activity, that is,
vinegar's hydronium ion activity is about 10−3
mol/L. In a similar vein, the decibel (symbol dB) is a unit of measure which is the
base-10 logarithm of ratios, such as power levels and voltage levels. It is mostly used in telecommunication, electronics,
and acoustics. In spectrometry and optics, the absorbance unit used to measure optical density is equivalent to −10 dB.
The Richter scale measures earthquake intensity on a base-10 logarithmic scale.

13. A semi-logarithmic plot of cases and deaths in the 2009 outbreak of influenza A (H1N1).
Semilog graphs depict one, typically the vertical, axis using a logarithmic scaling. This way, exponential functions of the
form f(x) = a · bx
appear as a straight line whose slope is proportional to b. At the right, numbers of cases of the swine flu are
shown—the horizontal (time) axis is linear, with the dates evenly spaced, the vertical (cases) axis is logarithmic, with the evenly
spaced divisions being labelled with successive powers of two. In a similar vein, log-log graphs scale both axes logarithmically.
A function f(x) is said to grow logarithmically, if it is (sometimes approximately) proportional to the logarithm. An example is
the harmonic series:
grows about as fast as ln(n).
[edit]Psychology
In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation
(though Stevens' power law is typically more accurate). According to the logarithmic responsiveness of the eye to
brightness, the apparent magnitude measures the brightness ofstars logarithmically.
Mathematically untrained individuals tend to estimate numerals with a logarithmic spacing, i.e., the position of a
presented numeral correlates with the logarithm of the given number so that smaller numbers are given more space than
bigger ones. With increasing mathematical training this logarithmic representation becomes more and more linear, a
development that has been found both in Western school children (comparing second to sixth graders)[12]
as in
comparison between American and indigene cultures.[13]
[edit]Complexity and entropy
In computer science, logarithms often appear in bounds for computational complexity. For example, to sort N items using
the quicksort algorithm typically requires time proportional to the product N · log(N). Similarly, base-2 logarithms are
used to express the amount of storage space or memory required for a binary representation of a number—with k bits
(each a 0 or a 1) one can represent 2k
distinct values, so any natural number N can be represented in no more than
(log2 N) + 1 bits.

14. Using logarithms, the notion of entropy in information theory is a measure of quantity of information. If a message
recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by
any one such message is quantified as log2 N bits. In the same vein, the concept of entropy also appears
in thermodynamics. Fractal dimension and Hausdorff dimension measure how much space geometric structures occupy.
Example
point (straight) line Koch curve Sierpinski triangle plane Apollonian sphere
Hausdorff dimension 0 1 log(4)/log(3) ≈ 1.262 log(3)/log(2) ≈ 1.585 2 ≈ 2.474
[edit]Mathematics
Graph comparing π(x) (red), x / ln x(green) and Li(x) (blue).
Natural logarithms have a tendency to appear in number theory, more specifically in counting primes. For any given
number x, the number of prime numbers less than or equal to x is denoted π(x). In its simples form, the prime number
theorem asserts that π(x) is approximately given by
in the sense that the ratio of π(x) and that fraction approaches 1 when x tends to infinity. This can be rephrased
by saying that the probability that a randomly chosen number between 1 and x is prime is indirectly proportional to
the numbers of decimal digits of x. A far better estimate of π(x) is given by the offset logarithmic integral function
Li(x), defined by

15. The Riemann hypothesis, one of the oldest open mathematical conjectures, can be stated in terms of comparing π(x) and Li(x).
The Erdős–Kac theorem describing the number of distinct prime factors also involves the natural logarithm.
In geometry the logarithm is used to form the metric for the half-plane model of hyperbolic geometry.
[edit]Statistics
In inferential statistics, the logarithm of the data in a dataset can be used for parametric statistical testing if the original data do
not meet the assumption of normality.
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 pl
ay
Just
major
third
play
Major third play Tritone play
Octa
ve
p
lay
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
400 600
120
0
[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]

16. 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
potentials.
The double or iterated logarithm, ln(ln(x)), is the inverse function of the double exponential function. The super- or hyper-4-
logarithm is the inverse function of tetration. The super-logarithm of x grows even more slowly than the double logarithm for
large x.
The Lambert W function is the inverse function of ƒ(w) = wew
. Polylogarithm is a generalization of the logarithm defined by
For s = 1, it is related to the logarithm via Li1(z) = −ln(1 − z). Forz = 1, on the other hand, Lis(1) yields the Riemann zeta
function ζ(s).
From the pure mathematical perspective, the identity log(cd) = log(c) + log(d) expresses an isomorphism between
the multiplicative group of the positive real numbers and the group of all the reals under addition. Logarithmic functions are the
only continuous isomorphisms from the multiplicative group of positive real numbers to the additive group of real numbers. The
logarithm function can be extended to aHaar measure in the topological group of positive real numbers under multiplication.
The discrete logarithm is a related notion in the theory of finite groups. It involves solving the equation bn
= x, where b and x are
elements of the group, and n is an integer specifying a power in the group operation. For some finite groups, it is believed that
the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications
in public key cryptography.[18]
Logarithms can be defined for p-adic numbers,[19]
quaternions and octonions. The logarithm of a matrix is the inverse of
the matrix exponential.
[

17. A more modern definition and explanation from 1866 A Dictionary of Science, Literature, & Art: Comprising the
Definitions and Derivations of the Scientific Terms in General Use, together with the History and Descriptions of
the Scientific Principles of Nearly Every Branch of Human Knowledge
Michael Stifel published Arithmetica integra in Nuremberg in 1544 which contains his
discovery of logarithms.
The method of logarithms was publicly propounded in 1614, in a book entitled Mirifici
Logarithmorum Canonis Descriptio, by John Napier, Baron of Merchiston, in Scotland,
[20]
(Joost Bürgi independently discovered logarithms; however, he did not publish his
discovery until four years after Napier). Early resistance to the use of logarithms was muted
by Kepler's enthusiastic support and his publication of a clear and impeccable explanation of
how they worked.[21]
Their use contributed to the advance of science, and especially of astronomy, by making
some difficult calculations possible. Prior to the advent of calculators and computers, they

18. were used constantly in surveying, navigation, and other branches of practical mathematics.
It supplanted the more involved method of prosthaphaeresis, which relied on trigonometric
identities as a quick method of computing products. Besides the utility of the logarithm
concept in computation, the natural logarithm presented a solution to the problem
of quadrature of a hyperbolic sector at the hand ofGregoire de Saint-Vincent in 1647.
At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers".
Later, Napier formed the word logarithm to mean a number that indicates a
ratio: λόγος (logos) meaning proportion, and ριθμόςἀ (arithmos) meaning number. Napier
chose that because the difference of two logarithms determines the ratio of the numbers they
represent, so that an arithmetic series of logarithms corresponds to a geometric series of
numbers. The term antilogarithm was introduced in the late 17th century and, while never
used extensively in mathematics, persisted in collections of tables until they fell into disuse.
Napier did not use a base as we now understand it, but his logarithms were, up to a scaling
factor, effectively to base 1/e. For interpolation purposes and ease of calculation, it is useful
to make the ratio r in the geometric series close to 1. Napier chose r = 1 − 10−7
= 0.999999
(Bürgi choser = 1 + 10−4
= 1.0001). Napier's original logarithms did not have log 1 = 0 but
rather log 107
= 0. Thus if N is a number and L is its logarithm as calculated by
Napier, N = 107
(1 − 10−7
) L
. Since (1 − 10−7
)107
is approximately 1/e, this
makes L / 107
approximately equal to log1/e N/107
.[9]
Part of a 20th-century table of common logarithms in the reference bookAbramowitz and Stegun.

19. Page of a 1912 table, which was common in schools.
[edit]Tables of logarithms
Main article: Table of logarithms
Prior to the advent of computers and calculators, using logarithms for practical purposes was
done with tables of logarithms, which contain both logarithms and antilogarithms with respect
to some fixed base.
When the purpose was to facilitate arithmetic computations, base 10 was used, and
logarithms of numbers between 1 and 10 by small increments (e.g. 0.01 or 0.001) appeared.
There was no need for logarithms of other numbers (above 10 or below 1) because moving
the decimal point corresponds to simply adding an integer to the logarithm. See Common
logarithm for further details.
The following table lists the major tables of logarithms compiled in history: