Natural logarithm of 2 explained

The decimal value of the natural logarithm of 2 is approximately

ln2 ≈ 0.693147180559945309417232121458.

The logarithm of 2 in other bases is obtained with the formula

log_b2=

	ln2
	lnb

The common logarithm in particular is

log₁₀2 ≈ 0.301029995663981195.

The inverse of this number is the binary logarithm of 10:

log₂10=

	1
	log₁₀2

≈ 3.321928095

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.

Series representations

Rising alternate factorial

ln2=

	infty
\sum
	n=1

	(-1)ⁿ⁺¹	=1-
	n

12+

13-

14+

15-	16+ … .

This is the well-known "alternating harmonic series".

ln2=

	1	+
	2

	1
	2

	infty
\sum
	n=1

	(-1)ⁿ⁺¹
	n(n+1)

ln2=

	5	+
	8

	1
	2

	infty
\sum
	n=1

	(-1)ⁿ⁺¹
	n(n+1)(n+2)

ln2=

	2	+
	3

	3
	4

	infty
\sum
	n=1

	(-1)ⁿ⁺¹
	n(n+1)(n+2)(n+3)

ln2=

	131	+
	192

	3
	2

	infty
\sum
	n=1

	(-1)ⁿ⁺¹
	n(n+1)(n+2)(n+3)(n+4)

ln2=

	661	+
	960

	15
	4

	infty
\sum
	n=1

	(-1)ⁿ⁺¹
	n(n+1)(n+2)(n+3)(n+4)(n+5)

ln2=

	2	.(1+
	3

	2	+
	4^3-4

	2	+
	8^3-8

	2
	12^3-12

+.........).

Binary rising constant factorial

ln2=

	infty
\sum
	n=1

	1
	2ⁿn

ln2=1

	infty
-\sum
	n=1

	1
	2ⁿn(n+1)

ln2=

	1
	2

	infty
\sum
	n=1

	1
	2ⁿn(n+1)(n+2)

ln2=

	5
	6

-6

	infty
\sum
	n=1

	1
	2ⁿn(n+1)(n+2)(n+3)

ln2=

	7
	12

+24

	infty
\sum
	n=1

	1
	2ⁿn(n+1)(n+2)(n+3)(n+4)

ln2=

	47
	60

-120

	infty
\sum
	n=1

	1
	2ⁿn(n+1)(n+2)(n+3)(n+4)(n+5)

Other series representations

	infty
\sum
	n=0

	1
	(2n+1)(2n+2)

=ln2.

	infty
\sum
	n=1

	1
	n(4n^2-1)

=2ln2-1.

	infty
\sum
	n=1

	(-1)ⁿ
	n(4n^2-1)

=ln2-1.

	infty
\sum
	n=1

	(-1)ⁿ
	n(9n^2-1)

=2ln2-

	3
	2

	infty
\sum
	n=1

	1
	4n^2-2n

=ln2.

	infty
\sum
	n=1

	2(-1)ⁿ⁺¹(2n-1)+1
	8n^2-4n

=ln2.

	infty
\sum
	n=0

	(-1)ⁿ
	3n+1

	ln2	+
	3

	\pi
	3\sqrt{3

	infty
\sum
	n=0

	(-1)ⁿ
	3n+2

	ln2	+
	3

	\pi
	3\sqrt{3

	infty
\sum
	n=0

	(-1)ⁿ
	(3n+1)(3n+2)

	2ln2
	3

	infty
\sum
	n=1

	n
\sum		k²
	k=1

=18-24ln2

using

\lim_{N →}

	2N
\sum
	n=N

	1
	n

=ln2

	infty
\sum
	n=1

	1
	4n^2-3n

=ln2+

	\pi
	6

(sums of the reciprocals of decagonal numbers)

Involving the Riemann Zeta function

	infty
\sum
	n=1

	1
	n

[\zeta(2n)-1]=ln2.

	infty
\sum
	n=2

	1
	2ⁿ

[\zeta(n)-1]=ln2-

	1
	2

	infty
\sum
	n=1

	1
	2n+1

[\zeta(2n+1)-1]=1-\gamma-

	ln2
	2

	infty
\sum
	n=1

	1
	2^2n-1(2n+1)

\zeta(2n)=1-ln2.

(is the Euler–Mascheroni constant and Riemann's zeta function.)

BBP-type representations

ln2=

	2
	3

	1
	2

	infty
\sum		\left(
	k=1

	1	+
	2k

	1	+
	4k+1

	1	+
	8k+4

	1
	16k+12

\right)

	1
	16^k

(See more about Bailey–Borwein–Plouffe (BBP)-type representations.)

Applying the three general series for natural logarithm to 2 directly gives:

ln2=

	infty
\sum
	n=1

	(-1)^n-1
	n

ln2=

	infty
\sum
	n=1

	1
	2ⁿn

ln2=

	2
	3

	infty
\sum
	k=0

	1
	9^k(2k+1)

Applying them to

style2=

	3
	2

⋅

	4
	3

gives:

ln2=

	infty
\sum
	n=1

	(-1)^n-1
	2ⁿn

	infty
\sum
	n=1

	(-1)^n-1
	3ⁿn

ln2=

	infty
\sum
	n=1

	1
	3ⁿn

	infty
\sum
	n=1

	1
	4ⁿn

ln2=

	2
	5

	infty
\sum
	k=0

	1
	25^k(2k+1)

	2
	7

	infty
\sum
	k=0

	1
	49^k(2k+1)

Applying them to

style2=(\sqrt{2})²

gives:

ln2=2

	infty
\sum
	n=1

	(-1)^n-1
	(\sqrt{2

+1)ⁿn}.

ln2=2

	infty
\sum
	n=1

	1
	(2+\sqrt{2

)ⁿn}.

ln2=

	4
	3+2\sqrt{2

} \sum_^\infty \frac .

Applying them to

style2={\left(

	16
	15

\right)}⁷ ⋅ {\left(

	81
	80

\right)}³ ⋅ {\left(

	25
	24

\right)}⁵

gives:

ln2=7

	infty
\sum
	n=1

	(-1)^n-1
	15ⁿn

	infty
\sum
	n=1

	(-1)^n-1
	80ⁿn

	infty
\sum
	n=1

	(-1)^n-1
	24ⁿn

ln2=7

	infty
\sum
	n=1

	1
	16ⁿn

	infty
\sum
	n=1

	1
	81ⁿn

	infty
\sum
	n=1

	1
	25ⁿn

ln2=

	14
	31

	infty
\sum
	k=0

	1
	961^k(2k+1)

	6
	161

	infty
\sum
	k=0

	1
	25921^k(2k+1)

	10
	49

	infty
\sum
	k=0

	1
	2401^k(2k+1)

Representation as integrals

The natural logarithm of 2 occurs frequently as the result of integration. Some explicit formulas for it include:

	1
\int
	0

	dx
	1+x

	2
\int
	1

	dx
	x

=ln2

	infty
\int
	0

e^-x

	1-e^-x
	x

dx=ln2

	infty
\int
	0

2^-xdx=

	1
	ln2

	\pi
	3

\int

\tanx

	\pi
	4

dx=2\int

\tanxdx=ln2

-	1
	\pii

	infty
\int
	0

	lnxlnlnx
	(x+1)²

dx=ln2

Other representations

The Pierce expansion is

ln2=1-

	1	+
	1 ⋅ 3

	1
	1 ⋅ 3 ⋅ 12

- … .

The Engel expansion is

ln2=

	1
	2

	1
	2 ⋅ 3

	1
	2 ⋅ 3 ⋅ 7

	1
	2 ⋅ 3 ⋅ 7 ⋅ 9

+ … .

The cotangent expansion is

ln2=\cot({\arccot(0)-\arccot(1)+\arccot(5)-\arccot(55)+\arccot(14187)- … }).

The simple continued fraction expansion is

ln2=\left[0;1,2,3,1,6,3,1,1,2,1,1,1,1,3,10,1,1,1,2,1,1,1,1,3,2,3,1,...\right]

,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:

ln2=\left[0;1,2,3,1,5,\tfrac{2}{3},7,\tfrac{1}{2},9,\tfrac{2}{5},...,2k-1,

	2
	k

,...\right]

,^[1]

also expressible as

ln2=\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{2}{2+\cfrac{2}{5+\cfrac{3}{2+\cfrac{3}{7+\cfrac{4}{2+\ddots}}}}}}}} =\cfrac{2}{3-\cfrac{1^2}{9-\cfrac{2^2}{15-\cfrac{3^2}{21-\ddots}}}}

Bootstrapping other logarithms

Given a value of, 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 based on their factorizations

c=2ⁱ³^j5^k7^{l … →}ln(c)=iln(2)+jln(3)+kln(5)+lln(7)+ …

This employs

In a third layer, the logarithms of rational numbers are computed with, and logarithms of roots via .

The logarithm of 2 is useful in the sense that the powers of 2 are rather densely distributed; finding powers close to powers of other numbers is comparatively easy, and series representations of are found by coupling 2 to with logarithmic conversions.

Example

If with some small, then and therefore

slnp-tlnq=ln\left(1+

	d
	q^t

\right)=

	infty
\sum
	m=1

(-1)^m+1

(	d	)^m
	q^t

	infty
\sum
	n=0

	2	{\left(
	2n+1

	d
	2q^t+d

\right)}²ⁿ⁺¹.

Selecting represents by and a series of a parameter that one wishes to keep small for quick convergence. Taking, for example, generates

2ln3=3ln2-\sum_k\ge

	(-1)^k
	8^kk

=3ln2+

	infty
\sum
	n=0

	2	{\left(
	2n+1

	1
	2 ⋅ 8+1

\right)}²ⁿ⁺¹.

This is actually the third line in the following table of expansions of this type:


1	3	1	2	= …
1	3	2	2	− = −…
2	3	3	2	= …
5	3	8	2	− = −…
12	3	19	2	= …
1	5	2	2	= …
3	5	7	2	− = −…
1	7	2	2	= …
1	7	3	2	− = −…
5	7	14	2	= …
1	11	3	2	= …
2	11	7	2	− = −…
11	11	38	2	= …
1	13	3	2	= …
1	13	4	2	− = −…
3	13	11	2	= …
7	13	26	2	− = −…
10	13	37	2	= …
1	17	4	2	= …
1	19	4	2	= …
4	19	17	2	− = −…
1	23	4	2	= …
1	23	5	2	− = −…
2	23	9	2	= …
1	29	4	2	= …
1	29	5	2	− = −…
7	29	34	2	= …
1	31	5	2	− = −…
1	37	5	2	= …
4	37	21	2	− = −…
5	37	26	2	= …
1	41	5	2	= …
2	41	11	2	− = −…
3	41	16	2	= …
1	43	5	2	= …
2	43	11	2	− = −…
5	43	27	2	= …
7	43	38	2	− = −…

Starting from the natural logarithm of one might use these parameters:


10	2	3	10	= …
21	3	10	10	= …
3	5	2	10	= …
10	5	7	10	− = −…
6	7	5	10	= …
13	7	11	10	− = −…
1	11	1	10	= …
1	13	1	10	= …
8	13	9	10	− = −…
9	13	10	10	= …
1	17	1	10	= …
4	17	5	10	− = −…
9	17	11	10	= …
3	19	4	10	− = −…
4	19	5	10	= …
7	19	9	10	− = −…
2	23	3	10	− = −…
3	23	4	10	= …
2	29	3	10	− = −…
2	31	3	10	− = −…

Known digits

This is a table of recent records in calculating digits of . 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

References

Richard P.. Brent. Fast multiple-precision evaluation of elementary functions. J. ACM. 23. 2. 1976. 242–251. 10.1145/321941.321944. 0395314. 6761843. free.
Horace S.. Uhler. Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17. Proc. Natl. Acad. Sci. U.S.A.. 26. 3. 1940. 205–212. 0001523. 10.1073/pnas.26.3.205. 1078033. 16588339. 1940PNAS...26..205U. free.
Dura W.. Sweeney. On the computation of Euler's constant. Mathematics of Computation. 1963. 17. 82. 170–178. 10.1090/S0025-5718-1963-0160308-X . 0160308. free.
Marc. Chamberland. Binary BBP-formulae for logarithms and generalized Gaussian–Mersenne primes. Journal of Integer Sequences. 6. 03.3.7. 2003. 2003JIntS...6...37C. 2046407. 2010-04-29. https://web.archive.org/web/20110606014342/http://www.emis.de/journals/JIS/VOL6/Chamberland/chamberland60.pdf. 2011-06-06. dead.
Boris. Gourévitch. Jesús. Guillera Goyanes. Construction of binomial sums for and polylogarithmic constants inspired by BBP formulas. Applied Math. E-Notes. 7. 2007. 2346048. 237–246.
Qiang. Wu. On the linear independence measure of logarithms of rational numbers. Mathematics of Computation. 72. 242. 901–911. 10.1090/S0025-5718-02-01442-4. 2003. free.

External links

- Web site: The logarithm constant:log 2. Gourdon. Xavier. Sebah. Pascal.

Notes and References

2004. 13. J. . Borwein. R. . Crandall. G. . Free. Exper. Math. . On the Ramanujan AGM Fraction, I: The Real-Parameter Case. 3 . 278–280. 10.1080/10586458.2004.10504540. 17758274.
Web site: y-cruncher. numberworld.org. 10 December 2018.
Web site: Natural log of 2. numberworld.org. 10 December 2018.
Web site: Records set by y-cruncher. https://web.archive.org/web/20200915070251/http://www.numberworld.org/y-cruncher/. September 15, 2020. 2020-09-15.
Web site: Natural logarithm of 2 (Log(2)) world record by Seungmin Kim. 19 August 2020. September 15, 2020.
Web site: Records set by y-cruncher. October 26, 2021.
Web site: Natural Log of 2 - William Echols. October 26, 2021.

Natural logarithm of 2 explained

Series representations

Rising alternate factorial

Binary rising constant factorial

Other series representations

Involving the Riemann Zeta function

BBP-type representations

Representation as integrals

Other representations

Bootstrapping other logarithms

Example

Known digits

See also

References

External links

Notes and References