The decimal value of the natural logarithm of 2 (sequence A002162 in the OEIS ) is approximately
The logarithm of 2 in other bases is obtained with the formula
The common logarithm in particular is ( OEIS: A007524 )
The inverse of this number is the binary logarithm of 10:
By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number.
(γ is the Euler–Mascheroni constant and ζ Riemann's zeta function.)
(See more about Bailey–Borwein–Plouffe (BBP)-type representations.)
Applying the three general series for natural logarithm to 2 directly gives:
Applying them to gives:
Applying them to gives:
Applying them to gives:
The natural logarithm of 2 occurs frequently as the result of integration. Some explicit formulas for it include:
The Pierce expansion is OEIS: A091846
The Engel expansion is OEIS: A059180
The cotangent expansion is OEIS: A081785
The simple continued fraction expansion is OEIS: A016730
which yields rational approximations, the first few of which are 0, 1, 2/3, 7/10, 9/13 and 61/88.
This generalized continued fraction:
Given a value of ln 2, a scheme of computing the logarithms of other integers is to tabulate the logarithms of the prime numbers and in the next layer the logarithms of the composite numbers c based on their factorizations
This employs
prime | approximate natural logarithm | OEIS |
---|---|---|
2 | 0.693147180559945309417232121458 | A002162 |
3 | 1.09861228866810969139524523692 | A002391 |
5 | 1.60943791243410037460075933323 | A016628 |
7 | 1.94591014905531330510535274344 | A016630 |
11 | 2.39789527279837054406194357797 | A016634 |
13 | 2.56494935746153673605348744157 | A016636 |
17 | 2.83321334405621608024953461787 | A016640 |
19 | 2.94443897916644046000902743189 | A016642 |
23 | 3.13549421592914969080675283181 | A016646 |
29 | 3.36729582998647402718327203236 | A016652 |
31 | 3.43398720448514624592916432454 | A016654 |
37 | 3.61091791264422444436809567103 | A016660 |
41 | 3.71357206670430780386676337304 | A016664 |
43 | 3.76120011569356242347284251335 | A016666 |
47 | 3.85014760171005858682095066977 | A016670 |
53 | 3.97029191355212183414446913903 | A016676 |
59 | 4.07753744390571945061605037372 | A016682 |
61 | 4.11087386417331124875138910343 | A016684 |
67 | 4.20469261939096605967007199636 | A016690 |
71 | 4.26267987704131542132945453251 | A016694 |
73 | 4.29045944114839112909210885744 | A016696 |
79 | 4.36944785246702149417294554148 | A016702 |
83 | 4.41884060779659792347547222329 | A016706 |
89 | 4.48863636973213983831781554067 | A016712 |
97 | 4.57471097850338282211672162170 | A016720 |
In a third layer, the logarithms of rational numbers r = a/b are computed with ln(r) = ln(a) − ln(b), and logarithms of roots via ln n√c = 1/n ln(c).
The logarithm of 2 is useful in the sense that the powers of 2 are rather densely distributed; finding powers 2i close to powers bj of other numbers b is comparatively easy, and series representations of ln(b) are found by coupling 2 to b with logarithmic conversions.
If ps = qt + d with some small d, then ps/qt = 1 + d/qt and therefore
Selecting q = 2 represents ln p by ln 2 and a series of a parameter d/qt that one wishes to keep small for quick convergence. Taking 32 = 23 + 1, for example, generates
This is actually the third line in the following table of expansions of this type:
s | p | t | q | d/qt |
---|---|---|---|---|
1 | 3 | 1 | 2 | 1/2 = 0.50000000… |
1 | 3 | 2 | 2 | −1/4 = −0.25000000… |
2 | 3 | 3 | 2 | 1/8 = 0.12500000… |
5 | 3 | 8 | 2 | −13/256 = −0.05078125… |
12 | 3 | 19 | 2 | 7153/524288 = 0.01364326… |
1 | 5 | 2 | 2 | 1/4 = 0.25000000… |
3 | 5 | 7 | 2 | −3/128 = −0.02343750… |
1 | 7 | 2 | 2 | 3/4 = 0.75000000… |
1 | 7 | 3 | 2 | −1/8 = −0.12500000… |
5 | 7 | 14 | 2 | 423/16384 = 0.02581787… |
1 | 11 | 3 | 2 | 3/8 = 0.37500000… |
2 | 11 | 7 | 2 | −7/128 = −0.05468750… |
11 | 11 | 38 | 2 | 10433763667/274877906944 = 0.03795781… |
1 | 13 | 3 | 2 | 5/8 = 0.62500000… |
1 | 13 | 4 | 2 | −3/16 = −0.18750000… |
3 | 13 | 11 | 2 | 149/2048 = 0.07275391… |
7 | 13 | 26 | 2 | −4360347/67108864 = −0.06497423… |
10 | 13 | 37 | 2 | 419538377/137438953472 = 0.00305254… |
1 | 17 | 4 | 2 | 1/16 = 0.06250000… |
1 | 19 | 4 | 2 | 3/16 = 0.18750000… |
4 | 19 | 17 | 2 | −751/131072 = −0.00572968… |
1 | 23 | 4 | 2 | 7/16 = 0.43750000… |
1 | 23 | 5 | 2 | −9/32 = −0.28125000… |
2 | 23 | 9 | 2 | 17/512 = 0.03320312… |
1 | 29 | 4 | 2 | 13/16 = 0.81250000… |
1 | 29 | 5 | 2 | −3/32 = −0.09375000… |
7 | 29 | 34 | 2 | 70007125/17179869184 = 0.00407495… |
1 | 31 | 5 | 2 | −1/32 = −0.03125000… |
1 | 37 | 5 | 2 | 5/32 = 0.15625000… |
4 | 37 | 21 | 2 | −222991/2097152 = −0.10633039… |
5 | 37 | 26 | 2 | 2235093/67108864 = 0.03330548… |
1 | 41 | 5 | 2 | 9/32 = 0.28125000… |
2 | 41 | 11 | 2 | −367/2048 = −0.17919922… |
3 | 41 | 16 | 2 | 3385/65536 = 0.05165100… |
1 | 43 | 5 | 2 | 11/32 = 0.34375000… |
2 | 43 | 11 | 2 | −199/2048 = −0.09716797… |
5 | 43 | 27 | 2 | 12790715/134217728 = 0.09529825… |
7 | 43 | 38 | 2 | −3059295837/274877906944 = −0.01112965… |
Starting from the natural logarithm of q = 10 one might use these parameters:
s | p | t | q | d/qt |
---|---|---|---|---|
10 | 2 | 3 | 10 | 3/125 = 0.02400000… |
21 | 3 | 10 | 10 | 460353203/10000000000 = 0.04603532… |
3 | 5 | 2 | 10 | 1/4 = 0.25000000… |
10 | 5 | 7 | 10 | −3/128 = −0.02343750… |
6 | 7 | 5 | 10 | 17649/100000 = 0.17649000… |
13 | 7 | 11 | 10 | −3110989593/100000000000 = −0.03110990… |
1 | 11 | 1 | 10 | 1/10 = 0.10000000… |
1 | 13 | 1 | 10 | 3/10 = 0.30000000… |
8 | 13 | 9 | 10 | −184269279/1000000000 = −0.18426928… |
9 | 13 | 10 | 10 | 604499373/10000000000 = 0.06044994… |
1 | 17 | 1 | 10 | 7/10 = 0.70000000… |
4 | 17 | 5 | 10 | −16479/100000 = −0.16479000… |
9 | 17 | 11 | 10 | 18587876497/100000000000 = 0.18587876… |
3 | 19 | 4 | 10 | −3141/10000 = −0.31410000… |
4 | 19 | 5 | 10 | 30321/100000 = 0.30321000… |
7 | 19 | 9 | 10 | −106128261/1000000000 = −0.10612826… |
2 | 23 | 3 | 10 | −471/1000 = −0.47100000… |
3 | 23 | 4 | 10 | 2167/10000 = 0.21670000… |
2 | 29 | 3 | 10 | −159/1000 = −0.15900000… |
2 | 31 | 3 | 10 | −39/1000 = −0.03900000… |
This is a table of recent records in calculating digits of ln 2. As of December 2018, it has been calculated to more digits than any other natural logarithm [2] [3] of a natural number, except that of 1.
Date | Name | Number of digits |
---|---|---|
January 7, 2009 | A.Yee & R.Chan | 15,500,000,000 |
February 4, 2009 | A.Yee & R.Chan | 31,026,000,000 |
February 21, 2011 | Alexander Yee | 50,000,000,050 |
May 14, 2011 | Shigeru Kondo | 100,000,000,000 |
February 28, 2014 | Shigeru Kondo | 200,000,000,050 |
July 12, 2015 | Ron Watkins | 250,000,000,000 |
January 30, 2016 | Ron Watkins | 350,000,000,000 |
April 18, 2016 | Ron Watkins | 500,000,000,000 |
December 10, 2018 | Michael Kwok | 600,000,000,000 |
April 26, 2019 | Jacob Riffee | 1,000,000,000,000 |
August 19, 2020 | Seungmin Kim [4] [5] | 1,200,000,000,100 |
September 9, 2021 | William Echols [6] [7] | 1,500,000,000,000 |
The exponential function is a mathematical function denoted by or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number, but various modern definitions allow it to be rigorously extended to all real arguments , including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".
In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n,
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, logex, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as
In mathematics, Catalan's constantG, is defined by
In mathematics, the Euler numbers are a sequence En of integers defined by the Taylor series expansion
In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value .
In mathematics, Stirling's approximation is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of . It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre.
In mathematics, the error function, often denoted by erf, is a function defined as:
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
In mathematics, the Wallis product for π, published in 1656 by John Wallis, states that
In mathematics, the lemniscate constantϖ is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of π for the circle. Equivalently, the perimeter of the lemniscate is 2ϖ. The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755. The symbol ϖ is a cursive variant of π; see Pi § Variant pi.
In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted and is named after the mathematician Bernhard Riemann. When the argument is a real number greater than one, the zeta function satisfies the equation
In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar-.
Approximations for the mathematical constant pi in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century.
Carl Johan Malmsten was a Swedish mathematician and politician. He is notable for early research into the theory of functions of a complex variable, for the evaluation of several important logarithmic integrals and series, for his studies in the theory of Zeta-function related series and integrals, as well as for helping Mittag-Leffler start the journal Acta Mathematica. Malmsten became Docent in 1840, and then, Professor of mathematics at the Uppsala University in 1842. He was elected a member of the Royal Swedish Academy of Sciences in 1844. He was also a minister without portfolio in 1859–1866 and Governor of Skaraborg County in 1866–1879.