Log4 Log25

Exponent of a power of two

In mathematics, the
binary logarithm
(log_iin
) is the power to which the number
two
must be raised to obtain the valuen. That is, for any existent number
10,

$$


x
=

log

2


⁡


due north

⟺




2

x


=
northward
.


{\displaystyle x=\log _{2}due north\quad \Longleftrightarrow \quad 2^{ten}=n.}

For example, the binary logarithm of
1
is
, the binary logarithm of
2
is
1, the binary logarithm of
iv
is2, and the binary logarithm of
32
is5.

The binary logarithm is the logarithm to the base of operations
ii
and is the inverse function of the ability of two function. Also as
log_two
, an alternative notation for the binary logarithm is
lb
(the notation preferred by ISO 31-11 and ISO 80000-2).

Historically, the first application of binary logarithms was in music theory, past Leonhard Euler: the binary logarithm of a frequency ratio of two musical tones gives the number of octaves past which the tones differ. Binary logarithms can be used to calculate the length of the representation of a number in the binary numeral system, or the number of bits needed to encode a message in data theory. In computer science, they count the number of steps needed for binary search and related algorithms. Other areas in which the binary logarithm is frequently used include combinatorics, bioinformatics, the design of sports tournaments, and photography.

Binary logarithms are included in the standard C mathematical functions and other mathematical software packages. The integer part of a binary logarithm tin be plant using the discover first set operation on an integer value, or by looking up the exponent of a floating point value. The partial part of the logarithm tin be calculated efficiently.

History

[edit]

The powers of 2 have been known since antiquity; for example, they appear in Euclid’southward
Elements, Props. Ix.32 (on the factorization of powers of ii) and IX.36 (one-half of the Euclid–Euler theorem, on the structure of even perfect numbers). And the binary logarithm of a power of two is simply its position in the ordered sequence of powers of two. On this basis, Michael Stifel has been credited with publishing the first known table of binary logarithms in 1544. His volume
Arithmetica Integra
contains several tables that prove the integers with their corresponding powers of ii. Reversing the rows of these tables allow them to be interpreted every bit tables of binary logarithms.^[1]
[ii]

Earlier than Stifel, the 8th century Jain mathematician Virasena is credited with a precursor to the binary logarithm. Virasena’s concept of
ardhacheda
has been defined as the number of times a given number can exist divided evenly past 2. This definition gives rise to a function that coincides with the binary logarithm on the powers of 2,^[3]
merely it is different for other integers, giving the 2-adic order rather than the logarithm.^[4]

The modern course of a binary logarithm, applying to whatever number (not but powers of two) was considered explicitly past Leonhard Euler in 1739. Euler established the application of binary logarithms to music theory, long before their applications in information theory and reckoner science became known. As role of his piece of work in this area, Euler published a table of binary logarithms of the integers from 1 to 8, to 7 decimal digits of accuracy.^{[5]
[half dozen]}

Definition and backdrop

[edit]

The binary logarithm role may be defined every bit the inverse function to the power of two function, which is a strictly increasing role over the positive real numbers and therefore has a unique changed.^[7]
Alternatively, it may be defined as
ln
due north/ln ii, where
ln
is the natural logarithm, defined in any of its standard ways. Using the circuitous logarithm in this definition allows the binary logarithm to be extended to the complex numbers.^[8]

As with other logarithms, the binary logarithm obeys the following equations, which can be used to simplify formulas that combine binary logarithms with multiplication or exponentiation:^[9]

$$



log

2


⁡


x
y
=

log

2


⁡


x
+

log

2


⁡


y


{\displaystyle \log _{2}xy=\log _{ii}x+\log _{2}y}

$$



log

2


⁡




ten
y


=

log

two


⁡


x
−



log

2


⁡


y


{\displaystyle \log _{2}{\frac {10}{y}}=\log _{2}ten-\log _{2}y}

$$



log

2


⁡



ten

y


=
y

log

2


⁡


10
.


{\displaystyle \log _{two}x^{y}=y\log _{2}10.}

For more than, see listing of logarithmic identities.

Note

[edit]

In mathematics, the binary logarithm of a number
n
is often written as
log₂north
.^[10]
However, several other notations for this part accept been used or proposed, especially in awarding areas.

Some authors write the binary logarithm as
lg
due north
,^[11]
[12]
the notation listed in
The Chicago Manual of Style.^[13]
Donald Knuth credits this notation to a suggestion of Edward Reingold,^[fourteen]
only its use in both information theory and reckoner science dates to earlier Reingold was active.^[xv]
[16]
The binary logarithm has also been written as
log
n

with a prior argument that the default base of operations for the logarithm isii.^{[17]
[18]
[nineteen]}
Another notation that is often used for the same function (especially in the German language scientific literature) is
ld
n
,^{[20]
[21]
[22]}
from Latin
logarithmus dualis
^[twenty]
or
logarithmus dyadis.^[20]
The DIN 1302 [de], ISO 31-11 and ISO 80000-ii standards recommend yet another annotation,
lb
north
. Co-ordinate to these standards,
lg
n

should not exist used for the binary logarithm, as it is instead reserved for the mutual logarithm
log₁₀
n
.^{[23]
[24]
[25]}

Applications

[edit]

Data theory

[edit]

The number of digits (bits) in the binary representation of a positive integer
northward
is the integral part of
1 + log₂n
, i.e.^[12]

$$


⌊



log

2


⁡


n
⌋


+
1.


{\displaystyle \lfloor \log _{two}n\rfloor +i.}

In information theory, the definition of the amount of self-information and information entropy is often expressed with the binary logarithm, corresponding to making the bit the primal unit of measurement of data. With these units, the Shannon–Hartley theorem expresses the information capacity of a channel every bit the binary logarithm of its indicate-to-dissonance ratio, plus i. Nevertheless, the natural logarithm and the nat are likewise used in alternative notations for these definitions.^[26]

Combinatorics

[edit]

Although the natural logarithm is more important than the binary logarithm in many areas of pure mathematics such as number theory and mathematical analysis,^[27]
the binary logarithm has several applications in combinatorics:

• Every binary tree with
  n
  leaves has height at least
  log₂due north
  , with equality when
  north
  is a power of two and the tree is a consummate binary tree.^[28]
  Relatedly, the Strahler number of a river system with
  due north
  tributary streams is at nigh
  log₂northward
  + 1.^[29]
• Every family of sets with
  n
  unlike sets has at least
  log₂n
  
  elements in its union, with equality when the family is a ability ready.^[thirty]
• Every partial cube with
  n
  vertices has isometric dimension at least
  log_twon
  , and has at most
  
  
  1
  /
  ii
  
  north
  log₂northward
  
  edges, with equality when the partial cube is a hypercube graph.^[31]
• Co-ordinate to Ramsey’s theorem, every
  n-vertex undirected graph has either a clique or an contained set up of size logarithmic in
  n. The precise size that can be guaranteed is non known, but the all-time premises known on its size involve binary logarithms. In particular, all graphs accept a clique or independent ready of size at to the lowest degree
  
  
  1
  /
  ii
  
  log₂n
  (ane −
  o(1))
  and almost all graphs do not take a clique or contained set of size larger than
  2 log₂north
  (one +
  o(1)).^[32]
• From a mathematical analysis of the Gilbert–Shannon–Reeds model of random shuffles, 1 tin can evidence that the number of times one needs to shuffle an
  due north-carte du jour deck of cards, using riffle shuffles, to go a distribution on permutations that is close to uniformly random, is approximately
  
  
  3
  /
  ii
  
  log₂n
  . This calculation forms the ground for a recommendation that 52-card decks should be shuffled seven times.^[33]

Computational complexity

[edit]

The binary logarithm also frequently appears in the analysis of algorithms,^[nineteen]
not only because of the frequent employ of binary number arithmetic in algorithms, only also because binary logarithms occur in the analysis of algorithms based on two-style branching.^[14]
If a problem initially has
n
choices for its solution, and each iteration of the algorithm reduces the number of choices by a factor of 2, then the number of iterations needed to select a single pick is again the integral part of
log_iinorthward
. This idea is used in the analysis of several algorithms and data structures. For example, in binary search, the size of the problem to exist solved is halved with each iteration, and therefore roughly
log₂due north

iterations are needed to obtain a solution for a problem of size
due north.^[34]
Similarly, a perfectly counterbalanced binary search tree containing
n
elements has summit
log₂(north
+ one) − one.^[35]

The running fourth dimension of an algorithm is ordinarily expressed in big O annotation, which is used to simplify expressions past omitting their constant factors and lower-social club terms. Because logarithms in different bases differ from each other simply past a abiding factor, algorithms that run in

O(log₂n)
time tin also be said to run in, say,

O(log₁₃
north)
time. The base of the logarithm in expressions such as

O(log
n)
or

O(northward
log
n)
is therefore not important and can exist omitted.^[11]
[36]
However, for logarithms that appear in the exponent of a time bound, the base of the logarithm cannot exist omitted. For example,

O(two^log₂n
)
is not the same as

O(2^ln
n
)
because the former is equal to

O(north)
and the latter to

O(n
^0.6931…).

Algorithms with running fourth dimension

O(n logn)
are sometimes called linearithmic.^[37]
Some examples of algorithms with running time

O(log
n)
or

O(n
log
northward)
are:

• Average time quicksort and other comparison sort algorithms^[38]
• Searching in balanced binary search trees^[39]
• Exponentiation by squaring^[xl]
• Longest increasing subsequence^[41]

Binary logarithms also occur in the exponents of the time bounds for some split up and conquer algorithms, such equally the Karatsuba algorithm for multiplying
northward-bit numbers in time

O(due north
^log₂ 3),^[42]
and the Strassen algorithm for multiplying

n
×
n

matrices in fourth dimension

O(n
^{log₂ seven}).^[43]
The occurrence of binary logarithms in these running times tin can be explained by reference to the chief theorem for divide-and-conquer recurrences.

Bioinformatics

[edit]

In bioinformatics, microarrays are used to measure out how strongly different genes are expressed in a sample of biological material. Different rates of expression of a factor are often compared by using the binary logarithm of the ratio of expression rates: the log ratio of two expression rates is divers as the binary logarithm of the ratio of the ii rates. Binary logarithms allow for a user-friendly comparison of expression rates: a doubled expression rate tin be described by a log ratio of
ane, a halved expression rate tin can be described past a log ratio of
−1, and an unchanged expression charge per unit tin can exist described past a log ratio of cypher, for instance.^[44]

Data points obtained in this way are often visualized as a scatterplot in which one or both of the coordinate axes are binary logarithms of intensity ratios, or in visualizations such as the MA plot and RA plot that rotate and scale these log ratio scatterplots.^[45]

Music theory

[edit]

In music theory, the interval or perceptual difference betwixt ii tones is adamant past the ratio of their frequencies. Intervals coming from rational number ratios with small numerators and denominators are perceived as especially euphonious. The simplest and near important of these intervals is the octave, a frequency ratio of
2:1. The number of octaves by which two tones differ is the binary logarithm of their frequency ratio.^[46]

To study tuning systems and other aspects of music theory that crave finer distinctions between tones, it is helpful to have a measure of the size of an interval that is finer than an octave and is additive (every bit logarithms are) rather than multiplicative (every bit frequency ratios are). That is, if tones
x,
y, and
z
form a rising sequence of tones, then the mensurate of the interval from
10
to
y
plus the measure of the interval from
y
to
z
should equal the measure of the interval from
10
to
z. Such a measure is given by the cent, which divides the octave into
1200
equal intervals (12
semitones of
100
cents each). Mathematically, given tones with frequencies

f
_i

and

f
₂
, the number of cents in the interval from

f
₁

to

f
₂

is^[46]

$$



|

1200

log

two


⁡





f

1



f

2





|

.


{\displaystyle \left|1200\log _{2}{\frac {f_{1}}{f_{two}}}\right|.}

The millioctave is defined in the same way, but with a multiplier of
thou
instead of
1200.^[47]

Sports scheduling

[edit]

In competitive games and sports involving two players or teams in each game or friction match, the binary logarithm indicates the number of rounds necessary in a unmarried-elimination tournament required to determine a winner. For example, a tournament of
four
players requires
log_ii iv = 2
rounds to make up one’s mind the winner, a tournament of
32
teams requires
log_two 32 = 5
rounds, etc. In this case, for
n
players/teams where
n
is not a power of 2,
log₂n

is rounded up since it is necessary to have at least one round in which not all remaining competitors play. For example,
log₂ vi
is approximately
ii.585, which rounds upwardly to
3, indicating that a tournament of
six
teams requires
iii
rounds (either two teams sit out the first circular, or one team sits out the second round). The same number of rounds is as well necessary to determine a clear winner in a Swiss-system tournament.^[48]

Photography

[edit]

In photography, exposure values are measured in terms of the binary logarithm of the amount of light reaching the film or sensor, in accordance with the Weber–Fechner police describing a logarithmic response of the human visual organization to light. A single stop of exposure is one unit on a base-2
logarithmic scale.^[49]
[50]
More than precisely, the exposure value of a photograph is divers as

$$



log

2


⁡





N

2


t




{\displaystyle \log _{ii}{\frac {N^{ii}}{t}}}

where
North
is the f-number measuring the discontinuity of the lens during the exposure, and
t
is the number of seconds of exposure.^[51]

Binary logarithms (expressed as stops) are also used in densitometry, to express the dynamic range of calorie-free-sensitive materials or digital sensors.^[52]

Calculation

[edit]

Conversion from other bases

[edit]

An easy way to calculate
log₂n

on calculators that do not have a
log₂

role is to use the natural logarithm (ln) or the common logarithm (log
or
log₁₀
) functions, which are found on most scientific calculators. The specific change of logarithm base of operations formulae for this are:^[l]
[53]

$$



log

2


⁡


due north
=



ln
⁡


due north


ln
⁡


2



=




log

ten


⁡


n



log

x


⁡


2



,


{\displaystyle \log _{two}northward={\frac {\ln n}{\ln ii}}={\frac {\log _{x}northward}{\log _{10}2}},}

or approximately

$$



log

2


⁡


n
≈


1.442695
ln
⁡


n
≈


3.321928

log

x


⁡


n
.


{\displaystyle \log _{2}north\approx 1.442695\ln n\approx iii.321928\log _{ten}northward.}

Integer rounding

[edit]

The binary logarithm tin be fabricated into a role from integers and to integers past rounding it up or down. These 2 forms of integer binary logarithm are related by this formula:

$$


⌊



log

two


⁡


(
due north
)
⌋


=
⌈



log

2


⁡


(
northward
+
one
)
⌉


−


1
,

 if

north
≥


1.


{\displaystyle \lfloor \log _{2}(n)\rfloor =\lceil \log _{two}(n+ane)\rceil -ane,{\text{ if }}n\geq i.}

^[54]

The definition tin can be extended by defining

$$


⌊



log

ii


⁡


(

)
⌋


=
−


1


{\displaystyle \lfloor \log _{two}(0)\rfloor =-i}

. Extended in this way, this function is related to the number of leading zeros of the 32-flake unsigned binary representation of
x,
nlz(x).

$$


⌊



log

2


⁡


(
n
)
⌋


=
31
−


nlz
⁡


(
n
)
.


{\displaystyle \lfloor \log _{2}(north)\rfloor =31-\operatorname {nlz} (n).}

^[54]

The integer binary logarithm can be interpreted as the cipher-based index of the about significant
1
flake in the input. In this sense it is the complement of the find first set operation, which finds the index of the to the lowest degree significant
1
bit. Many hardware platforms include support for finding the number of leading zeros, or equivalent operations, which can be used to rapidly find the binary logarithm. The
fls
and
flsl
functions in the Linux kernel^[55]
and in some versions of the libc software library also compute the binary logarithm (rounded upward to an integer, plus i).

Iterative approximation

[edit]

For a general positive existent number, the binary logarithm may exist computed in ii parts.^[56]
Start, one computes the integer part,

$$


⌊



log

2


⁡


ten
⌋




{\displaystyle \lfloor \log _{2}10\rfloor }

(called the characteristic of the logarithm). This reduces the problem to one where the argument of the logarithm is in a restricted range, the interval
[ane, 2), simplifying the 2nd stride of computing the fractional part (the mantissa of the logarithm). For whatsoever

x
> 0, there exists a unique integer
n
such that
ii
ⁿ

≤
x
< 2
ⁿ⁺¹
, or equivalently
1 ≤ 2^−north

x
< 2. Now the integer part of the logarithm is just
due north, and the fractional part is
log₂(two^−north

ten).^[56]
In other words:

$$



log

2


⁡


x
=
n
+

log

ii


⁡


y


where

y
=

2

−


northward


x

 and

y
∈


[
one
,
2
)


{\displaystyle \log _{two}x=n+\log _{2}y\quad {\text{where }}y=ii^{-due north}x{\text{ and }}y\in [i,2)}

For normalized floating-point numbers, the integer office is given by the floating-point exponent,^[57]
and for integers it can be adamant past performing a count leading zeros functioning.^[58]

The partial function of the result is
log₂y

and tin can be computed iteratively, using only elementary multiplication and division.^[56]
The algorithm for computing the partial role can be described in pseudocode as follows:

1. Offset with a real number
   y
   in the half-open up interval
   [1, two). If
   
   y
   = 1, so the algorithm is done, and the fractional office is zero.
2. Otherwise, square
   y
   repeatedly until the consequence
   z
   lies in the interval
   [2, 4). Let
   g
   be the number of squarings needed. That is,
   
   z
   =
   y
   ^{2
   thou}
   
   
   
   with
   1000
   chosen such that
   z
   is in
   [two, 4).
3. Taking the logarithm of both sides and doing some algebra:

   
   
   
   
   
   
   
   
   log
   
   2
   
   
   ⁡
   
   
   z
   
   
   
   =
   
   2
   
   thousand
   
   
   
   log
   
   ii
   
   
   ⁡
   
   
   y
   
   
   
   
   
   log
   
   two
   
   
   ⁡
   
   
   y
   
   
   
   =
   
   
   
   
   log
   
   two
   
   
   ⁡
   
   
   z
   
   
   two
   
   m
   
   
   
   
   
   
   
   
   
   
   =
   
   
   
   1
   +
   
   log
   
   2
   
   
   ⁡
   
   
   (
   z
   
   /
   
   two
   )
   
   
   two
   
   one thousand
   
   
   
   
   
   
   
   
   
   
   =
   
   2
   
   −
   
   
   thousand
   
   
   +
   
   2
   
   −
   
   
   thousand
   
   
   
   log
   
   two
   
   
   ⁡
   
   
   (
   z
   
   /
   
   2
   )
   .
   
   
   
   
   
   
   {\displaystyle {\begin{aligned}\log _{two}z&=2^{thou}\log _{ii}y\\\log _{two}y&={\frac {\log _{two}z}{ii^{m}}}\\&={\frac {1+\log _{2}(z/ii)}{2^{thou}}}\\&=2^{-m}+2^{-m}\log _{2}(z/2).\stop{aligned}}}
   
   

   

4. Over again
   
   z/two
   is a real number in the interval
   [one, 2). Return to step 1 and compute the binary logarithm of
   
   z/ii
   using the same method.

The consequence of this is expressed by the following recursive formulas, in which

$$



m

i




{\displaystyle m_{i}}

is the number of squarings required in the
i-th iteration of the algorithm:

In the special case where the fractional office in step ane is institute to be zippo, this is a
finite
sequence terminating at some indicate. Otherwise, it is an infinite serial that converges co-ordinate to the ratio test, since each term is strictly less than the previous 1 (since every

m

_i

> 0). For practical use, this infinite series must exist truncated to reach an approximate result. If the series is truncated after the
i-thursday term, then the error in the outcome is less than
2<sup>−(1000
_one
+
m
_ii
+ ⋯ +
m

_i
)</sup>
.^[56]

Software library back up

[edit]

The
log2
function is included in the standard C mathematical functions. The default version of this office takes double precision arguments just variants of information technology allow the argument to be single-precision or to be a long double.^[59]
In computing environments supporting circuitous numbers and implicit blazon conversion such equally MATLAB the argument to the
log2
function is immune to be a negative number, returning a circuitous one.^[60]

References

[edit]

External links

[edit]

• 
  Weisstein, Eric Westward., “Binary Logarithm”,
  MathWorld
  
  
  
• Anderson, Sean Eron (December 12, 2003), “Observe the log base two of an North-fleck integer in O(lg(N)) operations”,
  Bit Twiddling Hacks, Stanford University, retrieved
  2015-xi-25
  
  
  
• Feynman and the Connectedness Machine