2 Log 1 2

Exponent of a power of ii

In mathematics, the
binary logarithm
(log₂n
) is the power to which the number
2
must be raised to obtain the valuenorthward. That is, for any real number
x,

$$


x
=

log

2


⁡


due north

⟺




2

ten


=
northward
.


{\displaystyle 10=\log _{ii}n\quad \Longleftrightarrow \quad 2^{x}=n.}

For example, the binary logarithm of
ane
is
, the binary logarithm of
ii
is
i, the binary logarithm of
four
is2, and the binary logarithm of
32
is5.

The binary logarithm is the logarithm to the base
2
and is the inverse function of the power of two function. Also as
log₂
, an alternative annotation for the binary logarithm is
lb
(the notation preferred past 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 exist used to summate the length of the representation of a number in the binary numeral arrangement, or the number of bits needed to encode a message in information theory. In figurer scientific discipline, 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 constitute using the discover showtime set operation on an integer value, or past looking up the exponent of a floating point value. The fractional office of the logarithm can be calculated efficiently.

History

[edit]

The powers of two have been known since antiquity; for instance, they appear in Euclid’s
Elements, Props. Nine.32 (on the factorization of powers of two) and Ix.36 (half of the Euclid–Euler theorem, on the structure of even perfect numbers). And the binary logarithm of a power of two is just 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 book
Arithmetica Integra
contains several tables that prove the integers with their respective powers of two. Reversing the rows of these tables allow them to be interpreted every bit tables of binary logarithms.^[one]
[two]

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 tin be divided evenly by 2. This definition gives ascension to a function that coincides with the binary logarithm on the powers of two,^[three]
but it is different for other integers, giving the 2-adic order rather than the logarithm.^[4]

The modern form of a binary logarithm, applying to any number (not just powers of two) was considered explicitly by Leonhard Euler in 1739. Euler established the application of binary logarithms to music theory, long before their applications in information theory and figurer science became known. Every bit part of his work in this area, Euler published a table of binary logarithms of the integers from one to eight, to seven decimal digits of accuracy.^[5]
[6]

Definition and properties

[edit]

The binary logarithm function may be defined equally the inverse part to the power of ii office, which is a strictly increasing function over the positive real numbers and therefore has a unique inverse.^[7]
Alternatively, it may be defined as
ln
northward/ln 2, where
ln
is the natural logarithm, defined in whatsoever of its standard ways. Using the complex logarithm in this definition allows the binary logarithm to be extended to the complex numbers.^[8]

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

$$



log

2


⁡


ten
y
=

log

two


⁡


ten
+

log

2


⁡


y


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

$$



log

2


⁡




10
y


=

log

ii


⁡


ten
−



log

ii


⁡


y


{\displaystyle \log _{ii}{\frac {x}{y}}=\log _{ii}x-\log _{2}y}

$$



log

ii


⁡



10

y


=
y

log

2


⁡


x
.


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

For more, see listing of logarithmic identities.

Notation

[edit]

In mathematics, the binary logarithm of a number
due north
is often written as
log₂due north
.^[10]
However, several other notations for this function have been used or proposed, particularly in application areas.

Some authors write the binary logarithm equally
lg
north
,^[11]
[12]
the annotation listed in
The Chicago Manual of Style.^[13]
Donald Knuth credits this notation to a suggestion of Edward Reingold,^[fourteen]
but its use in both information theory and computer scientific discipline dates to earlier Reingold was active.^[xv]
[16]
The binary logarithm has too been written every bit
log
n

with a prior statement that the default base for the logarithm is2.^{[17]
[18]
[nineteen]}
Another annotation that is often used for the aforementioned function (especially in the German scientific literature) is
ld
n
,^{[xx]
[21]
[22]}
from Latin
logarithmus dualis
^[20]
or
logarithmus dyadis.^[20]
The DIN 1302 [de], ISO 31-11 and ISO 80000-2 standards recommend nonetheless another notation,
lb
n
. Co-ordinate to these standards,
lg
due north

should not exist used for the binary logarithm, as it is instead reserved for the common logarithm
log_x
due north
.^{[23]
[24]
[25]}

Applications

[edit]

Data theory

[edit]

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

$$


⌊



log

2


⁡


due north
⌋


+
1.


{\displaystyle \lfloor \log _{ii}n\rfloor +one.}

In data theory, the definition of the amount of self-information and information entropy is often expressed with the binary logarithm, respective to making the bit the key unit of information. With these units, the Shannon–Hartley theorem expresses the information capacity of a aqueduct every bit the binary logarithm of its signal-to-noise ratio, plus one. Notwithstanding, the natural logarithm and the nat are besides used in culling notations for these definitions.^[26]

Combinatorics

[edit]

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

• Every binary tree with
  n
  leaves has height at least
  log_iin
  , with equality when
  northward
  is a power of two and the tree is a complete binary tree.^[28]
  Relatedly, the Strahler number of a river system with
  n
  tributary streams is at about
  log₂n
  + 1.^[29]
• Every family of sets with
  due north
  unlike sets has at least
  log_twon
  
  elements in its union, with equality when the family is a power gear up.^[30]
• Every partial cube with
  n
  vertices has isometric dimension at least
  log₂n
  , and has at almost
  
  
  1
  /
  2
  
  n
  log_iin
  
  edges, with equality when the partial cube is a hypercube graph.^[31]
• According to Ramsey’south theorem, every
  n-vertex undirected graph has either a clique or an independent set of size logarithmic in
  n. The precise size that can be guaranteed is non known, but the best bounds known on its size involve binary logarithms. In particular, all graphs have a clique or contained set of size at least
  
  
  1
  /
  2
  
  log₂due north
  (i −
  o(1))
  and about all graphs do non have a clique or independent set of size larger than
  2 log₂n
  (one +
  o(1)).^[32]
• From a mathematical analysis of the Gilbert–Shannon–Reeds model of random shuffles, 1 can evidence that the number of times one needs to shuffle an
  north-card deck of cards, using riffle shuffles, to become a distribution on permutations that is close to uniformly random, is approximately
  
  
  3
  /
  ii
  
  log_twon
  . This calculation forms the basis for a recommendation that 52-menu decks should be shuffled 7 times.^[33]

Computational complexity

[edit]

The binary logarithm also frequently appears in the assay of algorithms,^[xix]
not only because of the frequent apply of binary number arithmetics in algorithms, simply also because binary logarithms occur in the analysis of algorithms based on 2-fashion 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 cistron of two, and so the number of iterations needed to select a single choice is again the integral function of
log₂n
. This thought is used in the analysis of several algorithms and information structures. For example, in binary search, the size of the problem to be solved is halved with each iteration, and therefore roughly
log₂due north

iterations are needed to obtain a solution for a problem of size
n.^[34]
Similarly, a perfectly balanced binary search tree containing
due north
elements has superlative
log_two(n
+ ane) − 1.^[35]

The running time of an algorithm is commonly expressed in big O notation, which is used to simplify expressions past omitting their constant factors and lower-order terms. Because logarithms in unlike bases differ from each other only past a constant cistron, algorithms that run in

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

O(log₁₃
northward)
fourth dimension. The base of the logarithm in expressions such as

O(log
northward)
or

O(due north
log
north)
is therefore non important and can be omitted.^{[eleven]
[36]}
However, for logarithms that appear in the exponent of a time leap, the base of the logarithm cannot be omitted. For example,

O(ii^{log_twonorthward}
)
is not the same as

O(2^{ln
due north}
)
considering the former is equal to

O(n)
and the latter to

O(northward
^0.6931…).

Algorithms with running time

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

O(log
northward)
or

O(n
log
n)
are:

• Average fourth dimension quicksort and other comparing sort algorithms^[38]
• Searching in balanced binary search trees^[39]
• Exponentiation past squaring^[40]
• Longest increasing subsequence^[41]

Binary logarithms also occur in the exponents of the time bounds for some divide and conquer algorithms, such as the Karatsuba algorithm for multiplying
n-fleck numbers in time

O(due north
^{log_ii three}),^[42]
and the Strassen algorithm for multiplying

n
×
n

matrices in fourth dimension

O(n
^log₂ 7).^[43]
The occurrence of binary logarithms in these running times can be explained past reference to the master theorem for split-and-conquer recurrences.

Bioinformatics

[edit]

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

Data points obtained in this mode 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 equally 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 between two tones is adamant past the ratio of their frequencies. Intervals coming from rational number ratios with pocket-size numerators and denominators are perceived every bit specially euphonious. The simplest and near of import 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 report tuning systems and other aspects of music theory that require finer distinctions between tones, information technology is helpful to have a mensurate of the size of an interval that is finer than an octave and is additive (equally logarithms are) rather than multiplicative (as frequency ratios are). That is, if tones
ten,
y, and
z
grade a rising sequence of tones, and then the measure of the interval from
x
to
y
plus the mensurate of the interval from
y
to
z
should equal the measure of the interval from
x
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
₁

and

f
₂
, the number of cents in the interval from

f
₁

to

f
₂

is^[46]

$$



|

1200

log

2


⁡





f

1



f

2





|

.


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

The millioctave is defined in the same way, but with a multiplier of
1000
instead of
1200.^[47]

Sports scheduling

[edit]

In competitive games and sports involving two players or teams in each game or match, the binary logarithm indicates the number of rounds necessary in a single-emptying tournament required to determine a winner. For example, a tournament of
four
players requires
log_ii four = 2
rounds to make up one’s mind the winner, a tournament of
32
teams requires
log₂ 32 = v
rounds, etc. In this example, for
n
players/teams where
n
is non a ability of 2,
log₂n

is rounded upward since it is necessary to have at least one round in which non all remaining competitors play. For instance,
log₂ six
is approximately
2.585, which rounds upwards to
3, indicating that a tournament of
6
teams requires
iii
rounds (either two teams sit out the start circular, or one team sits out the 2d round). The same number of rounds is besides necessary to determine a clear winner in a Swiss-organization 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 constabulary describing a logarithmic response of the man visual system to low-cal. A single stop of exposure is one unit on a base-2
logarithmic scale.^[49]
[l]
More precisely, the exposure value of a photograph is divers as

$$



log

two


⁡





North

2


t




{\displaystyle \log _{2}{\frac {N^{2}}{t}}}

where
N
is the f-number measuring the aperture 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 limited the dynamic range of light-sensitive materials or digital sensors.^[52]

Calculation

[edit]

Conversion from other bases

[edit]

An easy mode to calculate
log₂due north

on calculators that do not have a
log_two

function 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 formulae for this are:^[50]
[53]

$$



log

2


⁡


n
=



ln
⁡


n


ln
⁡


2



=




log

10


⁡


n



log

x


⁡


2



,


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

or approximately

$$



log

2


⁡


n
≈


1.442695
ln
⁡


n
≈


3.321928

log

10


⁡


northward
.


{\displaystyle \log _{2}n\approx 1.442695\ln northward\approx 3.321928\log _{10}n.}

Integer rounding

[edit]

The binary logarithm can be made into a function from integers and to integers past rounding it up or downward. These two forms of integer binary logarithm are related by this formula:

$$


⌊



log

ii


⁡


(
due north
)
⌋


=
⌈



log

ii


⁡


(
north
+
1
)
⌉


−


1
,

 if

n
≥


1.


{\displaystyle \lfloor \log _{2}(n)\rfloor =\lceil \log _{ii}(n+one)\rceil -1,{\text{ if }}n\geq 1.}

^[54]

The definition can be extended by defining

$$


⌊



log

ii


⁡


(

)
⌋


=
−


one


{\displaystyle \lfloor \log _{2}(0)\rfloor =-1}

. Extended in this fashion, this function is related to the number of leading zeros of the 32-bit unsigned binary representation of
x,
nlz(10).

$$


⌊



log

2


⁡


(
n
)
⌋


=
31
−


nlz
⁡


(
n
)
.


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

^[54]

The integer binary logarithm tin can be interpreted every bit the zero-based index of the near significant
1
bit in the input. In this sense it is the complement of the find start set operation, which finds the index of the least significant
1
bit. Many hardware platforms include support for finding the number of leading zeros, or equivalent operations, which tin can be used to apace find the binary logarithm. The
fls
and
flsl
functions in the Linux kernel^[55]
and in some versions of the libc software library likewise compute the binary logarithm (rounded up to an integer, plus 1).

Iterative approximation

[edit]

For a full general positive existent number, the binary logarithm may be computed in two parts.^[56]
First, i computes the integer office,

$$


⌊



log

ii


⁡


10
⌋




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

(chosen the feature of the logarithm). This reduces the trouble to i where the statement of the logarithm is in a restricted range, the interval
[i, 2), simplifying the second step of calculating the fractional function (the mantissa of the logarithm). For any

x
> 0, there exists a unique integer
north
such that
two
^northward

≤
x
< 2
ⁿ⁺¹
, or equivalently
1 ≤ two⁻ⁿ

x
< 2. Now the integer part of the logarithm is simply
northward, and the fractional function is
log₂(2^−north

ten).^[56]
In other words:

$$



log

2


⁡


x
=
north
+

log

2


⁡


y


where

y
=

two

−


n


x

 and

y
∈


[
ane
,
2
)


{\displaystyle \log _{two}x=n+\log _{2}y\quad {\text{where }}y=2^{-northward}x{\text{ and }}y\in [1,2)}

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

The fractional role of the result is
log_iiy

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

1. Start with a real number
   y
   in the one-half-open up interval
   [1, 2). If
   
   y
   = ane, and then the algorithm is done, and the fractional part is zero.
2. Otherwise, square
   y
   repeatedly until the outcome
   z
   lies in the interval
   [two, four). Let
   m
   be the number of squarings needed. That is,
   
   z
   =
   y
   ^{2
   m}
   
   
   
   with
   m
   chosen such that
   z
   is in
   [2, 4).
3. Taking the logarithm of both sides and doing some algebra:

   
   
   
   
   
   
   
   
   log
   
   two
   
   
   ⁡
   
   
   z
   
   
   
   =
   
   2
   
   1000
   
   
   
   log
   
   two
   
   
   ⁡
   
   
   y
   
   
   
   
   
   log
   
   2
   
   
   ⁡
   
   
   y
   
   
   
   =
   
   
   
   
   log
   
   2
   
   
   ⁡
   
   
   z
   
   
   two
   
   one thousand
   
   
   
   
   
   
   
   
   
   
   =
   
   
   
   1
   +
   
   log
   
   ii
   
   
   ⁡
   
   
   (
   z
   
   /
   
   ii
   )
   
   
   2
   
   m
   
   
   
   
   
   
   
   
   
   
   =
   
   ii
   
   −
   
   
   m
   
   
   +
   
   2
   
   −
   
   
   m
   
   
   
   log
   
   ii
   
   
   ⁡
   
   
   (
   z
   
   /
   
   2
   )
   .
   
   
   
   
   
   
   {\displaystyle {\begin{aligned}\log _{2}z&=2^{m}\log _{two}y\\\log _{2}y&={\frac {\log _{2}z}{2^{m}}}\\&={\frac {1+\log _{2}(z/2)}{2^{one thousand}}}\\&=ii^{-m}+2^{-m}\log _{2}(z/2).\cease{aligned}}}
   
   

   

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

The upshot 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 partial function in step 1 is found to be zero, this is a
finite
sequence terminating at some point. Otherwise, information technology is an infinite series that converges according to the ratio examination, since each term is strictly less than the previous one (since every

m

_i

> 0). For applied use, this infinite series must be truncated to reach an estimate result. If the series is truncated after the
i-thursday term, so the fault in the result is less than
two<sup>−(m
₁
+
m
_two
+ ⋯ +
grand

_i
)</sup>
.^[56]

Software library support

[edit]

The
log2
function is included in the standard C mathematical functions. The default version of this function takes double precision arguments just variants of it permit the statement to be single-precision or to be a long double.^[59]
In computing environments supporting complex numbers and implicit type conversion such equally MATLAB the statement to the
log2
part is immune to exist a negative number, returning a complex one.^[60]

References

[edit]

External links

[edit]

• 
  Weisstein, Eric W., “Binary Logarithm”,
  MathWorld
  
  
  
• Anderson, Sean Eron (Dec 12, 2003), “Discover the log base 2 of an N-flake integer in O(lg(N)) operations”,
  Bit Twiddling Hacks, Stanford University, retrieved
  2015-11-25
  
  
  
• Feynman and the Connection Machine