2 Log 1 2.
In mathematics, the
binary logarithm
(log_{2}n
) 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
log_{2}
, an alternative annotation for the binary logarithm is
lb
(the notation preferred past ISO 3111 and ISO 800002).
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.
Daftar Isi:
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 2adic 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 postobit 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
log_{2}due 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 , ISO 3111 and ISO 800002 standards recommend nonetheless another notation,
lb
n
. Coordinate 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 + log_{2}n
, i.e.^{[12]}

$$
⌊
log
2
due north
⌋
+
1.
{\displaystyle \lfloor \log _{ii}n\rfloor +one.}
In data theory, the definition of the amount of selfinformation 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 signaltonoise 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_{ii}n
, 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
log_{2}n
+ 1.^{[29]}  Every family of sets with
due north
unlike sets has at least
log_{two}n
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
log_{2}n
, and has at almost
1
/
2
n
log_{ii}n
edges, with equality when the partial cube is a hypercube graph.^{[31]}  According to Ramsey’south theorem, every
nvertex 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
log_{2}due north
(i −
o(1))
and about all graphs do non have a clique or independent set of size larger than
2 log_{2}n
(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
northcard deck of cards, using riffle shuffles, to become a distribution on permutations that is close to uniformly random, is approximately
3
/
ii
log_{two}n
. This calculation forms the basis for a recommendation that 52menu 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 2fashion 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
log_{2}n
. 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
log_{2}due 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 lowerorder terms. Because logarithms in unlike bases differ from each other only past a constant cistron, algorithms that run in
O(log_{2}n)
time can also be said to run in, say,
O(log_{13}
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^{logtwonorthward
})
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
nfleck numbers in time
O(due north
^{logii 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 splitandconquer 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 pocketsize 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 \left1200\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 singleemptying 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
log_{2} 32 = v
rounds, etc. In this example, for
n
players/teams where
n
is non a ability of 2,
log_{2}n
is rounded upward since it is necessary to have at least one round in which non all remaining competitors play. For instance,
log_{2} 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 Swissorganization 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 lowcal. A single stop of exposure is one unit on a base2
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 fnumber 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 lightsensitive materials or digital sensors.^{[52]}
Calculation
[edit]
Conversion from other bases
[edit]
An easy mode to calculate
log_{2}due north
on calculators that do not have a
log_{two}
function is to use the natural logarithm (ln) or the common logarithm (log
or
log_{10}
) 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
. Extended in this fashion, this function is related to the number of leading zeros of the 32bit 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 zerobased 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,
(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
log_{2}(2^{−north
}
ten).^{[56]}
In other words:

$$
log
2
x
=
north
+
log
2
y
where
y
=
two
−
n
x
and
y
∈
[
ane
,
2
)
{\displaystyle \log _{two}x=n+\log _{2}y\quad {\text{where }}y=2^{northward}x{\text{ and }}y\in [1,2)}
For normalized floatingsignal numbers, the integer role is given past the floatingpoint exponent,^{[57]}
and for integers it can exist adamant by performing a count leading zeros operation.^{[58]}
The fractional role of the result is
log_{ii}y
and can be computed iteratively, using only elementary multiplication and division.^{[56]}
The algorithm for computing the fractional part can be described in pseudocode as follows:
 Start with a real number
y
in the onehalfopen up interval
[1, 2). If
y
= ane, and then the algorithm is done, and the fractional part is zero.  Otherwise, square
y
repeatedly until the outcome
z
lies in the interval
[two, four). Let
m
be the number of squarings needed. That is,
z
=
y
^{2 m }
with
m
chosen such that
z
is in
[2, 4).  Taking the logarithm of both sides and doing some algebra:
$$$$
log
two
z
=
2
1000
log
two
y
log
2
y
=
log
2
z
two
one thousand
=
1
+
log
ii
(
z
/
ii
)
2
m
=
ii
−
m
+
2
−
m
log
ii
(
z
/
2
)
.
{\displaystyle {\begin{aligned}\log _{2}z&=2^{m}\log _{two}y\\\log _{2}y&={\frac {\log _{2}z}{2^{m}}}\\&={\frac {1+\log _{2}(z/2)}{2^{one thousand}}}\\&=ii^{m}+2^{m}\log _{2}(z/2).\cease{aligned}}}
 Once again
z/2
is a real number in the interval
[1, 2). Return to step 1 and compute the binary logarithm of
z/2
using the same method.
The upshot of this is expressed by the following recursive formulas, in which
is the number of squarings required in the
ith iteration of the algorithm:
$$$$
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
ithursday term, so the fault in the result is less than
two^{−(m
1
+
m
two
+ ⋯ +
grand
i
)}
.^{[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 singleprecision 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]

^
Groza, Vivian Shaw; Shelley, Susanne M. (1972),
Precalculus mathematics, New York: Holt, Rinehart and Winston, p. 182, ISBN9780030776700
. 
^
Stifel, Michael (1544),
Arithmetica integra
(in Latin), p. 31
. A copy of the same table with two more entries appears on p. 237, and another copy extended to negative powers appears on p. 249b. 
^
Joseph, G. Grand. (2011),
The Crest of the Peacock: NotEuropean Roots of Mathematics
(third ed.), Princeton University Press, p. 352
. 
^
Run across, east.1000.,
Shparlinski, Igor (2013),
Cryptographic Applications of Analytic Number Theory: Complication Lower Bounds and Pseudorandomness, Progress in Informatics and Applied Logic, vol. 22, Birkhäuser, p. 35, ISBN978iii03488037iv
. 
^
Euler, Leonhard (1739), “Chapter Seven. De Variorum Intervallorum Receptis Appelationibus”,
Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae
(in Latin), Leningrad University, pp. 102–112
. 
^
Tegg, Thomas (1829), “Binary logarithms”,
London encyclopaedia; or, Universal dictionary of scientific discipline, art, literature and practical mechanics: comprising a popular view of the present state of noesis, Book 4, pp. 142–143
. 
^
Batschelet, E. (2012),
Introduction to Mathematics for Life Scientists, Springer, p. 128, ISBN978iii642960802
. 
^
For case, Microsoft Excel provides the
IMLOG2
function for complex binary logarithms: see
Bourg, David 1000. (2006),
Excel Scientific and Applied science Cookbook, O’Reilly Media, p. 232, ISBN978059655317three
. 
^
Kolman, Bernard; Shapiro, Arnold (1982), “11.4 Properties of Logarithms”,
Algebra for College Students, Academic Press, pp. 334–335, ISBN978148327121vii
. 
^
For case, this is the notation used in the
Encyclopedia of Mathematics
and
The Princeton Companion to Mathematics. 
^
^{ a }
^{ b }
Cormen, Thomas H.; Leiserson, Charles East.; Rivest, Ronald Fifty.; Stein, Clifford (2001) [1990],
Introduction to Algorithms
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^
^{ a }
^{ b }
Sedgewick, Robert; Wayne, Kevin Daniel (2011),
Algorithms, AddisonWesley Professional person, p. 185, ISBN978032157351iii
. 
^
The Chicago Manual of Mode
(25th ed.), University of Chicago Press, 2003, p. 530
. 
^
^{ a }
^{ b }
Knuth, Donald Due east. (1997),
Fundamental Algorithms, The Art of Computer Programming, vol. 1 (tertiary ed.), AddisonWesley Professional, ISBN978032163574seven
, p. 11. The same note was in the 1973 2nd edition of the same book (p. 23) but without the credit to Reingold. 
^
Trucco, Ernesto (1956), “A note on the information content of graphs”,
Bull. Math. Biophys.,
18
(two): 129–135, doi:10.1007/BF02477836, MR 0077919
. 
^
Mitchell, John N. (1962), “Figurer multiplication and sectionalisation using binary logarithms”,
IRE Transactions on Electronic Computers, EC11 (4): 512–517, doi:10.1109/TEC.1962.5219391
. 
^
Fiche, Georges; Hebuterne, Gerard (2013),
Mathematics for Engineers, John Wiley & Sons, p. 152, ISBN978ane118623336,
In the following, and unless otherwise stated, the notation
log
x
ever stands for the logarithm to the base
2
of
x
. 
^
Cover, Thomas Grand.; Thomas, Joy A. (2012),
Elements of Information Theory
(2nd ed.), John Wiley & Sons, p. 33, ISBN978i11858577i,
Unless otherwise specified, we volition accept all logarithms to base
2
. 
^
^{ a }
^{ b }
Goodrich, Michael T.; Tamassia, Roberto (2002),
Algorithm Design: Foundations, Assay, and Internet Examples, John Wiley & Sons, p. 23,
One of the interesting and sometimes even surprising aspects of the analysis of data structures and algorithms is the ubiquitous presence of logarithms … As is the custom in the computing literature, we omit writing the base of operations
b
of the logarithm when
b = two.

^
^{ a }
^{ b }
^{ c }
Tafel, Hans Jörg (1971),
Einführung in die digitale Datenverarbeitung
[Introduction to digital data processing] (in German), Munich: Carl Hanser Verlag, pp. xx–21, ISBN3446105697

^
Tietze, Ulrich; Schenk, Christoph (1999),
HalbleiterSchaltungstechnik
(in German language) (1st corrected reprint, 11th ed.), Springer Verlag, p. 1370, ISBN3540641920

^
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Origins and Foundations of Calculating: In Cooperation with Heinz Nixdorf MuseumsForum, Springer Scientific discipline & Business Media, p. 54, ISBN978three64202991ii
. 
^
For DIN 1302 see
Brockhaus Enzyklopädie in zwanzig Bänden
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. 
^
For ISO 3111 run into
Thompson, Ambler; Taylor, Barry One thousand (March 2008),
Guide for the Apply of the International Arrangement of Units (SI) — NIST Special Publication 811, 2008 Edition — Second Printing
(PDF), NIST, p. 33
. 
^
For ISO 80000ii see
“Quantities and units – Part 2: Mathematical signs and symbols to be used in the natural sciences and engineering”
(PDF),
International Standard ISO 800002
(1st ed.), December 1, 2009, Section 12, Exponential and logarithmic functions, p. 18
. 
^
Van der Lubbe, Jan C. A. (1997),
Information Theory, Cambridge University Printing, p. 3, ISBN9780521467605
. 
^
Stewart, Ian (2015),
Taming the Infinite, Quercus, p. 120, ISBN9781623654733,
in advanced mathematics and science the only logarithm of importance is the natural logarithm
. 
^
Leiss, Ernst L. (2006),
A Programmer’southward Companion to Algorithm Analysis, CRC Press, p. 28, ISBN978i42001170eight
. 
^
Devroye, Fifty.; Kruszewski, P. (1996), “On the Horton–Strahler number for random tries”,
RAIRO Informatique Théorique et Applications,
xxx
(5): 443–456, doi:10.1051/ita/1996300504431, MR 1435732
. 
^
Equivalently, a family with
k
distinct elements has at most
two^{ k }
distinct sets, with equality when it is a power prepare. 
^
Eppstein, David (2005), “The lattice dimension of a graph”,
European Journal of Combinatorics,
26
(5): 585–592, arXiv:cs.DS/0402028, doi:ten.1016/j.ejc.2004.05.001, MR 2127682, S2CID 7482443
. 
^
Graham, Ronald L.; Rothschild, Bruce Fifty.; Spencer, Joel H. (1980),
Ramsey Theory, WileyInterscience, p. 78
. 
^
Bayer, Dave; Diaconis, Persi (1992), “Trailing the dovetail shuffle to its lair”,
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External links
[edit]

Weisstein, Eric W., “Binary Logarithm”,
MathWorld

Anderson, Sean Eron (Dec 12, 2003), “Discover the log base 2 of an Nflake integer in O(lg(N)) operations”,
Bit Twiddling Hacks, Stanford University, retrieved
20151125
 Feynman and the Connection Machine
2 Log 1 2
Source: https://en.wikipedia.org/wiki/Binary_logarithm