View source for Natural logarithm of 2

The decimal value of the [[natural logarithm]] of [[2 (number)|2]] {{OEIS|A002162}}
is approximately
:<math>\ln(2) \approx 0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458.</math>
The logarithm of 2 in other bases is obtained with the [[Logarithmic identities|formula]]
:<math>\log_b(2) = \frac{\ln(2)}{\ln(b)}.</math>
The [[Briggsian logarithm|common logarithm]] in particular is ({{OEIS2C|A007524}})
:<math>\log_{10}(2) \approx 0.301\,029\,995\,663\,981\,195.</math>
The inverse of this number is the [[binary logarithm]] of 10:
:<math> \log_2(10)=\frac{1}{\log_{10}(2)} \approx 3.321\,928\,095</math> ({{OEIS2C|A020862}}).

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]].

Although not proven, empirical evidence suggests that ln(2) is a [[normal number]]. The decimal counts in the first 600 billion digits are as follows <ref>{{cite web|url=http://www.numberworld.org/y-cruncher/records/2018_12_10_log2.txt|title=Validation file|author=|date=|work=numberworld.org|accessdate=10 December 2018}}</ref>
{| class="wikitable"
! number!!digit frequency
|-
| [[0 (number)|0]] || {{val|59999962241}}
|-
| [[1 (number)|1]] || {{val|59999935189}}
|-
| [[2 (number)|2]] || {{val|59999701650}} 
|-
| [[3 (number)|3]] || {{val|59999883740}}
|-
| [[4 (number)|4]] || {{val|59999893049}}
|-
| [[5 (number)|5]] || {{val|59999774875}}
|-
| [[6 (number)|6]] || {{val|60000238666}}
|-
| [[7 (number)|7]] || {{val|60000373799}}
|-
| [[8 (number)|8]] || {{val|60000148375}}
|-
| [[9 (number)|9]] || {{val|60000088416}}
|}

==Series representations==

===Rising alternate factorial===
:<math>\ln(2) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}=1-\frac12+\frac13-\frac14+\frac15-\frac16+\cdots.</math> This is the well-known "[[Harmonic series (mathematics)#Alternating harmonic series|alternating harmonic series]]".
:<math>\ln(2) = \tfrac{1}{2} +\tfrac{1}{2}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)}.</math>
:<math>\ln(2) = \tfrac{5}{8} +\tfrac{1}{2}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)(n+2)}.</math>
:<math>\ln(2) = \tfrac{2}{3} +\tfrac{3}{4}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)(n+2)(n+3)}.</math>
:<math>\ln(2) = \tfrac{131}{192} +\tfrac{3}{2}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)(n+2)(n+3)(n+4)}.</math>
:<math>\ln(2) = \tfrac{661}{960} +\tfrac{15}{4}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)(n+2)(n+3)(n+4)(n+5)}.</math>

===Binary rising constant factorial===
:<math>\ln(2) = \sum_{n=1}^\infty \frac{1}{2^{n}n}.</math>
:<math>\ln(2) = 1 -\sum_{n=1}^\infty \frac{1}{2^{n}n(n+1)}.</math>
:<math>\ln(2) = \tfrac{1}{2} + 2 \sum_{n=1}^\infty \frac{1}{2^{n}n(n+1)(n+2)} .</math>
:<math>\ln(2) = \tfrac{5}{6} - 6 \sum_{n=1}^\infty \frac{1}{2^{n}n(n+1)(n+2)(n+3)} .</math>
:<math>\ln(2) = \tfrac{7}{12} + 24 \sum_{n=1}^\infty \frac{1}{2^{n}n(n+1)(n+2)(n+3)(n+4)} .</math>
:<math>\ln(2) = \tfrac{47}{60} - 120 \sum_{n=1}^\infty \frac{1}{2^{n}n(n+1)(n+2)(n+3)(n+4)(n+5)} .</math>

===Other series representations===
:<math>\sum_{n=0}^\infty \frac{1}{(2n+1)(2n+2)} = \ln(2).</math>
:<math>\sum_{n=1}^\infty \frac{1}{n(4n^2-1)} = 2\ln(2) -1.</math>
:<math>\sum_{n=1}^\infty \frac{(-1)^n}{n(4n^2-1)} = \ln(2) -1.</math>
:<math>\sum_{n=1}^\infty \frac{(-1)^n}{n(9n^2-1)} = 2\ln(2) -\tfrac{3}{2}.</math>
:<math>\sum_{n=1}^\infty \frac{1}{4n^2-2n} = \ln(2).</math>
:<math>\sum_{n=1}^\infty \frac{2(-1)^{n+1}(2n-1)+1}{8n^2-4n} = \ln(2).</math>
:<math>\sum_{n=0}^\infty \frac{(-1)^{n}}{3n+1} = \frac{\ln(2)}{3}+\frac{\pi}{3\sqrt{3}}.</math>
:<math>\sum_{n=0}^\infty \frac{(-1)^{n}}{3n+2} = -\frac{\ln(2)}{3}+\frac{\pi}{3\sqrt{3}}.</math>
:<math>\sum_{n=0}^\infty \frac{(-1)^{n}}{(3n+1)(3n+2)} = \frac{2\ln(2)}{3}.</math>
:<math>\sum_{n=1}^\infty \frac{1}{\sum_{k=1}^n k^2} = 18 - 24 \ln(2)</math> using <math>\lim_{N\rightarrow \infty} \sum_{n=N}^{2N} \frac{1}{n} = \ln(2)</math>
:<math>\sum_{n=1}^\infty \frac{1}{4k^2-3k} = \ln(2) + \frac{\pi}{6} </math> (sums of the reciprocals of [[decagonal number]]s)

===Involving the Riemann Zeta function===
:<math>\sum_{n=2}^\infty \frac{1}{2^n}[\zeta(n)-1] = \ln(2) -\tfrac{1}{2}.</math>
:<math>\sum_{n=2}^\infty \frac{1}{2n+1}[\zeta(n)-1] = 1-\gamma-\frac{\ln(2)}{2}.</math>
:<math>\sum_{n=1}^\infty \frac{1}{2^{2n-1}(2n+1)}\zeta(2n) = 1-\ln(2).</math>

({{math|''γ''}} is the [[Euler constant|Euler–Mascheroni constant]] and {{math|''ζ''}} [[Riemann zeta function|Riemann's zeta function]].)

===BBP-type representations===
:<math>\ln(2) = \tfrac{2}{3} + \tfrac{1}{2} \sum_{k = 1}^\infty \left(\frac{1}{2k}+\frac{1}{4k+1}+\frac{1}{8k+4}+\frac{1}{16k+12}\right) \frac{1}{16^k} .</math>
(See more about [[Bailey–Borwein–Plouffe formula|Bailey–Borwein–Plouffe (BBP)-type representations]].)

Applying the three general series for natural logarithm to 2 directly gives:
:<math>\ln(2) = \sum_{n = 1}^\infty \frac{(-1)^{n-1}}{n}.</math>
:<math>\ln(2) = \sum_{n = 1}^\infty \frac{1}{2^{n}n}.</math>
:<math>\ln(2) = \tfrac{2}{3} \sum_{k = 0}^\infty \frac{1}{9^{k}(2k+1)}.</math>

Applying them to <math>\textstyle 2 = \frac{3}{2} \cdot \frac{4}{3} </math> gives:
:<math>\ln(2) = \sum_{n = 1}^\infty \frac{(-1)^{n-1}}{2^n n} + \sum_{n = 1}^\infty \frac{(-1)^{n-1}}{3^n n} .</math>
:<math>\ln(2) = \sum_{n = 1}^\infty \frac{1}{3^n n} + \sum_{n = 1}^\infty \frac{1}{4^n n} .</math>
:<math>\ln(2) = \frac{2}{5} \sum_{k = 0}^\infty \frac{1}{25^{k}(2k+1)} + \frac{2}{7} \sum_{k = 0}^\infty \frac{1}{49^{k}(2k+1)} .</math>

Applying them to <math>\textstyle 2 = (\sqrt{2})^2 </math> gives:
:<math>\ln(2) = 2 \sum_{n = 1}^\infty \frac{(-1)^{n-1}}{(\sqrt{2} + 1)^n n} .</math>
:<math>\ln(2) = 2 \sum_{n = 1}^\infty \frac{1}{(2 + \sqrt{2})^n n} .</math>
:<math>\ln(2) = \frac{4}{3 + 2 \sqrt{2}} \sum_{k = 0}^\infty \frac{1}{(17 + 12 \sqrt{2})^{k}(2k+1)} .</math>

Applying them to <math>\textstyle 2 = { \left( \frac{16}{15} \right) }^{7} \cdot { \left( \frac{81}{80} \right) }^{3} \cdot { \left( \frac{25}{24} \right) }^{5} </math> gives:
:<math>\ln(2) = 7 \sum_{n = 1}^\infty \frac{(-1)^{n-1}}{15^n n} + 3 \sum_{n = 1}^\infty \frac{(-1)^{n-1}}{80^n n} +  5 \sum_{n = 1}^\infty \frac{(-1)^{n-1}}{24^n n} .</math>
:<math>\ln(2) = 7 \sum_{n = 1}^\infty \frac{1}{16^n n} + 3 \sum_{n = 1}^\infty \frac{1}{81^n n} +  5 \sum_{n = 1}^\infty \frac{1}{25^n n} .</math>
:<math>\ln(2) = \frac{14}{31} \sum_{k = 0}^\infty \frac{1}{961^{k}(2k+1)} \, + \, \frac{6}{161} \sum_{k = 0}^\infty \frac{1}{25921^{k}(2k+1)} \, + \, \frac{10}{49} \sum_{k = 0}^\infty \frac{1}{2401^{k}(2k+1)} .</math>

==Representation as integrals==

:<math>\int_0^1 \frac{dx}{1+x} = \ln(2),\text{ or, equivalently, }\int_1^2 \frac{dx}{x} = \ln(2).</math>

:<math>\int_1^\infty \frac{dx}{(1+x^2)(1+x)^2} = \frac{1-\ln(2)}{4}.</math>

:<math>\int_0^\infty \frac{dx}{1+e^{nx}} = \frac{\ln(2)}{n};
\int_0^\infty \frac{dx}{3+e^{nx}} = \frac{2\ln(2)}{3n}.</math>

:<math>\int_0^\infty \left(\frac{1}{e^x-1}-\frac{2}{e^{2x}-1}\right)\,dx=\ln(2).</math>

:<math>\int_0^\infty e^{-x}\frac{1-e^{-x}}{x} \, dx= \ln(2).</math>

:<math>\int_0^1 \ln\left(\frac{x^2-1}{x\ln(x)}\right) \, dx=-1+\ln(2)+\gamma.</math>

:<math>\int_0^\frac{\pi}{3} \tan(x) \, dx=2\int_0^\frac{\pi}{4} \tan(x) \, dx=\ln(2).</math>

:<math>\int_{-\frac{\pi}{4}}^\frac{\pi}{4} \ln\left(\sin(x)+\cos(x)\right)\,dx=-\frac{\pi \ln(2)}{4}.</math>

:<math>\int_0^1 x^2\ln(1+x)\,dx=\frac{2\ln(2)}{3}-\frac{5}{18}.</math>

:<math>\int_0^1 x\ln(1+x)\ln(1-x)\,dx=\tfrac{1}{4}-\ln(2).</math>

:<math>\int_0^1 x^3\ln(1+x)\ln(1-x)\,dx=\tfrac{13}{96}-\frac{2\ln(2)}{3}.</math>

:<math>\int_0^1 \frac{\ln x}{(1+x)^2}\,dx = -\ln(2).</math>

:<math>\int_0^1 \frac{\ln(1+x)-x}{x^2}\,dx=1-2\ln(2).</math>

:<math>\int_0^1 \frac{dx}{x(1-\ln(x))(1-2\ln(x))} = \ln(2).</math>

:<math>\int_1^\infty \frac{\ln\left(\ln(x)\right)}{x^3}\,dx = -\frac{\gamma+\ln(2)}{2}.</math>

({{math|''γ''}} is the [[Euler constant|Euler–Mascheroni constant]].)

==Other representations==
The Pierce expansion is {{OEIS2C|A091846}}
:<math> \ln(2) = 1 -\frac{1}{1\cdot 3}+\frac{1}{1\cdot 3\cdot 12} -\cdots. </math>
The [[Engel expansion]] is {{OEIS2C|A059180}}
:<math> \ln(2) = \frac{1}{2} + \frac{1}{2\cdot 3} + \frac{1}{2\cdot 3\cdot 7} + \frac{1}{2\cdot 3\cdot 7\cdot 9}+\cdots. </math>
The cotangent expansion is {{OEIS2C|A081785}}
:<math> \ln(2) = \cot({\arccot(0) -\arccot(1) + \arccot(5) - \arccot(55) + \arccot(14187) -\cdots}). </math>
The simple [[continued fraction]] expansion is {{OEIS2C|A016730}}
:<math> \ln(2) = \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]</math>,
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]]:
:<math> \ln(2) = \left[ 0;1,2,3,1,5,\tfrac{2}{3},7,\tfrac{1}{2},9,\tfrac{2}{5},...,2k-1,\frac{2}{k},...\right] </math>,<ref>{{cite journal|year=2004|volume=13|first1=J. |last1=Borwein| first2=R. | last2=Crandall| first3=G. | last3=Free|journal=Exper. Math. |title= On the Ramanujan AGM Fraction , I: The Real-Parameter Case|number=3 | pages=278–280|url=http://www.kurims.kyoto-u.ac.jp/EMIS/journals/EM/expmath/volumes/13/13.3/BorweinCrandallFee.pdf|doi=10.1080/10586458.2004.10504540}}</ref>
:also expressible as
:<math> \ln(2) = \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}}}} </math>

==Bootstrapping other logarithms==
Given a value of {{math|ln(2)}}, a scheme of computing the logarithms of other [[integer]]s is to tabulate the logarithms of the [[prime number]]s and in the next layer the logarithms of the [[Composite number|composite]] numbers {{math|''c''}} based on their [[Table of prime factors|factorizations]]
:<math>c=2^i3^j5^k7^l\cdots\rightarrow \ln(c)=i\ln(2)+j\ln(3)+k\ln(5)+l\ln(7)+\cdots</math>
This employs
{| class="wikitable sortable"
! prime!!approximate natural logarithm!![[OEIS]]
|-
| [[2 (number)|2]] || {{val|0.693147180559945309417232121458}} || {{OEIS link|A002162}}
|-
| [[3 (number)|3]] || {{val|1.09861228866810969139524523692}} || {{OEIS link|A002391}}
|-
| [[5 (number)|5]] || {{val|1.60943791243410037460075933323}} || {{OEIS link|A016628}}
|-
| [[7 (number)|7]] || {{val|1.94591014905531330510535274344}} || {{OEIS link|A016630}}
|-
| [[11 (number)|11]] || {{val|2.39789527279837054406194357797}} || {{OEIS link|A016634}}
|-
| [[13 (number)|13]] || {{val|2.56494935746153673605348744157}} || {{OEIS link|A016636}}
|-
| [[17 (number)|17]] || {{val|2.83321334405621608024953461787}} || {{OEIS link|A016640}}
|-
| [[19 (number)|19]] || {{val|2.94443897916644046000902743189}} || {{OEIS link|A016642}}
|-
| [[23 (number)|23]] || {{val|3.13549421592914969080675283181}} || {{OEIS link|A016646}}
|-
| [[29 (number)|29]] || {{val|3.36729582998647402718327203236}} || {{OEIS link|A016652}}
|-
| [[31 (number)|31]] || {{val|3.43398720448514624592916432454}} || {{OEIS link|A016654}}
|-
| [[37 (number)|37]] || {{val|3.61091791264422444436809567103}} || {{OEIS link|A016660}}
|-
| [[41 (number)|41]] || {{val|3.71357206670430780386676337304}} || {{OEIS link|A016664}}
|-
| [[43 (number)|43]] || {{val|3.76120011569356242347284251335}} || {{OEIS link|A016666}}
|-
| [[47 (number)|47]] || {{val|3.85014760171005858682095066977}} || {{OEIS link|A016670}}
|-
| [[53 (number)|53]] || {{val|3.97029191355212183414446913903}} || {{OEIS link|A016676}}
|-
| [[59 (number)|59]] || {{val|4.07753744390571945061605037372}} || {{OEIS link|A016682}}
|-
| [[61 (number)|61]] || {{val|4.11087386417331124875138910343}} || {{OEIS link|A016684}}
|-
| [[67 (number)|67]] || {{val|4.20469261939096605967007199636}} || {{OEIS link|A016690}}
|-
| [[71 (number)|71]] || {{val|4.26267987704131542132945453251}} || {{OEIS link|A016694}}
|-
| [[73 (number)|73]] || {{val|4.29045944114839112909210885744}} || {{OEIS link|A016696}}
|-
| [[79 (number)|79]] || {{val|4.36944785246702149417294554148}} || {{OEIS link|A016702}}
|-
| [[83 (number)|83]] || {{val|4.41884060779659792347547222329}} || {{OEIS link|A016706}}
|-
| [[89 (number)|89]] || {{val|4.48863636973213983831781554067}} || {{OEIS link|A016712}}
|-
| [[97 (number)|97]] || {{val|4.57471097850338282211672162170}} || {{OEIS link|A016720}}
|}

In a third layer, the logarithms of rational numbers {{math|''r'' {{=}} {{sfrac|''a''|''b''}}}} are computed with {{math|ln(''r'') {{=}} ln(''a'') − ln(''b'')}}, and logarithms of roots via {{math|ln {{radic|''c''|''n''}} {{=}} {{sfrac|1|''n''}} ln(''c'')}}.

The logarithm of [[2 (number)|2]] is useful in the sense that the powers of 2 are rather densely distributed; finding powers {{math|2{{sup|''i''}}}} close to powers {{math|''b{{sup|j}}''}} of other numbers {{math|''b''}} is comparatively easy, and series representations of {{math|ln(''b'')}} are found by coupling 2 to {{math|''b''}} with [[Logarithmic identities|logarithmic conversions]].

===Example===
If {{math|''p{{sup|s}}'' {{=}} ''q{{sup|t}}'' + ''d''}} with some small {{math|''d''}}, then {{math|{{sfrac|''p{{sup|s}}''|''q{{sup|t}}''}} {{=}} 1 + {{sfrac|''d''|''q{{sup|t}}''}}}} and therefore
:<math> s\ln(p)-t\ln(q) = \ln\left(1+\frac{d}{q^t}\right) = \sum_{m=1}^\infty (-1)^{m+1}\frac{(\frac{d}{q^t})^m}{m} = \sum_{n=0}^\infty \frac{2}{2n+1} {\left(\frac{d}{2 q^t + d}\right)}^{2n+1} .</math>
Selecting {{math|''q'' {{=}} 2}} represents {{math|ln(''p'')}} by {{math|ln(2)}} and a series of a parameter {{math|{{sfrac|''d''|''q{{sup|t}}''}}}} that one wishes to keep small for quick convergence. Taking {{math|3{{sup|2}} {{=}} 2{{sup|3}} + 1}}, for example, generates
:<math>2\ln(3) = 3\ln(2)-\sum_{k\ge 1}\frac{(-1)^k}{8^{k}k} = 3\ln(2) + \sum_{n=0}^\infty \frac{2}{2n+1} {\left(\frac{1}{2 \cdot 8 + 1}\right)}^{2n+1} .</math>
This is actually the third line in the following table of expansions of this type:
<!--
:<math> 5\ln(3) = 8\ln(2)-\sum_{k\ge 1}\frac{1}{k}(\frac{13}{256})^k.</math>
:<math> 6\ln(5) = 14\ln(2)-\sum_{k\ge 1}\frac{1}{k}(\frac{759}{16384})^k.</math>
:<math> \ln(7) = 3\ln(2)-\sum_{k\ge 1}\frac{1}{8^{k}k}.</math>
:<math> 5\ln(7) = 14\ln(2)-\sum_{k\ge 1}\frac{(-1)^k}{k}(\frac{423}{16384})^k.</math>
:<math> \ln(17) = 4\ln(2)-\sum_{k\ge 1}\frac{(-1)^k}{16^{k}k}.</math>
-->
{| class="wikitable sortable" style="text-align: right"
! {{math|''s''}}!!{{math|''p''}}!!{{math|''t''}}!!{{math|''q''}}!!{{math|{{sfrac|''d''|''q{{sup|t}}''}}}}
|-
| 1 || 3 || 1 || 2 || {{sfrac|1|2}} = {{0|−}}{{val|0.50000000}}…
|-
| 1 || 3 || 2 || 2 || −{{sfrac|1|4}} = −{{val|0.25000000}}…
|-
| 2 || 3 || 3 || 2 || {{sfrac|1|8}} = {{0|−}}{{val|0.12500000}}…
|-
| 5 || 3 || 8 || 2 || −{{sfrac|13|256}} = −{{val|0.05078125}}…
|-
| 12 || 3 || 19 || 2 || {{sfrac|7153|{{val|524288}}}} = {{0|−}}{{val|0.01364326}}…
|-
| 1 || 5 || 2 || 2 || {{sfrac|1|4}} = {{0|−}}{{val|0.25000000}}…
|-
| 3 || 5 || 7 || 2 || −{{sfrac|3|128}} = −{{val|0.02343750}}…
|-
| 1 || 7 || 2 || 2 || {{sfrac|3|4}} = {{0|−}}{{val|0.75000000}}…
|-
| 1 || 7 || 3 || 2 || −{{sfrac|1|8}} = −{{val|0.12500000}}…
|-
| 5 || 7 || 14 || 2 || {{sfrac|423|{{val|16384}}}} = {{0|−}}{{val|0.02581787}}…
|-
| 1 || 11 || 3 || 2 || {{sfrac|3|8}} = {{0|−}}{{val|0.37500000}}…
|-
| 2 || 11 || 7 || 2 || −{{sfrac|7|128}} = −{{val|0.05468750}}…
|-
| 11 || 11 || 38 || 2 || {{sfrac|{{val|10433763667}}|{{val|274877906944}}}} = {{0|−}}{{val|0.03795781}}…
|-
| 1 || 13 || 3 || 2 || {{sfrac|5|8}} = {{0|−}}{{val|0.62500000}}…
|-
| 1 || 13 || 4 || 2 || −{{sfrac|3|16}} = −{{val|0.18750000}}…
|-
| 3 || 13 || 11 || 2 || {{sfrac|149|2048}} = {{0|−}}{{val|0.07275391}}…
|-
| 7 || 13 || 26 || 2 || −{{sfrac|{{val|4360347}}|{{val|67108864}}}} = −{{val|0.06497423}}…
|-
| 10 || 13 || 37 || 2 || {{sfrac|{{val|419538377}}|{{val|137438953472}}}} = {{0|−}}{{val|0.00305254}}…
|-
| 1 || 17 || 4 || 2 || {{sfrac|1|16}} = {{0|−}}{{val|0.06250000}}…
|-
| 1 || 19 || 4 || 2 || {{sfrac|3|16}} = {{0|−}}{{val|0.18750000}}…
|-
| 4 || 19 || 17 || 2 || −{{sfrac|751|{{val|131072}}}} = −{{val|0.00572968}}…
|-
| 1 || 23 || 4 || 2 || {{sfrac|7|16}} = {{0|−}}{{val|0.43750000}}…
|-
| 1 || 23 || 5 || 2 || −{{sfrac|9|32}} = −{{val|0.28125000}}…
|-
| 2 || 23 || 9 || 2 || {{sfrac|17|512}} = {{0|−}}{{val|0.03320312}}…
|-
| 1 || 29 || 4 || 2 || {{sfrac|13|16}} = {{0|−}}{{val|0.81250000}}…
|-
| 1 || 29 || 5 || 2 || −{{sfrac|3|32}} = −{{val|0.09375000}}…
|-
| 7 || 29 || 34 || 2 || {{sfrac|{{val|70007125}}|{{val|17179869184}}}} = {{0|−}}{{val|0.00407495}}…
|-
| 1 || 31 || 5 || 2 || −{{sfrac|1|32}} = −{{val|0.03125000}}…
|-
| 1 || 37 || 5 || 2 || {{sfrac|5|32}} = {{0|−}}{{val|0.15625000}}…
|-
| 4 || 37 || 21 || 2 || −{{sfrac|{{val|222991}}|{{val|2097152}}}} = −{{val|0.10633039}}…
|-
| 5 || 37 || 26 || 2 || {{sfrac|{{val|2235093}}|{{val|67108864}}}} = {{0|−}}{{val|0.03330548}}…
|-
| 1 || 41 || 5 || 2 || {{sfrac|9|32}} = {{0|−}}{{val|0.28125000}}…
|-
| 2 || 41 || 11 || 2 || −{{sfrac|367|2048}} = −{{val|0.17919922}}…
|-
| 3 || 41 || 16 || 2 || {{sfrac|3385|{{val|65536}}}} = {{0|−}}{{val|0.05165100}}…
|-
| 1 || 43 || 5 || 2 || {{sfrac|11|32}} = {{0|−}}{{val|0.34375000}}…
|-
| 2 || 43 || 11 || 2 || −{{sfrac|199|2048}} = −{{val|0.09716797}}…
|-
| 5 || 43 || 27 || 2 || {{sfrac|{{val|12790715}}|{{val|134217728}}}} = {{0|−}}{{val|0.09529825}}…
|-
| 7 || 43 || 38 || 2 || −{{sfrac|{{val|3059295837}}|{{val|274877906944}}}} = −{{val|0.01112965}}…
|}

Starting from the natural logarithm of {{math|''q'' {{=}} 10}} one might use these parameters:
{| class="wikitable sortable" style="text-align: right"
! {{math|''s''}}!!{{math|''p''}}!!{{math|''t''}}!!{{math|''q''}}!!{{math|{{sfrac|''d''|''q{{sup|t}}''}}}}
|-
| 10 || 2 || 3 || 10 || {{sfrac|3|125}} = {{0|−}}{{val|0.02400000}}…
|-
| 21 || 3 || 10 || 10 || {{sfrac|{{val|460353203}}|{{val|10000000000}}}} = {{0|−}}{{val|0.04603532}}…
|-
| 3 || 5 || 2 || 10 || {{sfrac|1|4}} = {{0|−}}{{val|0.25000000}}…
|-
| 10 || 5 || 7 || 10 || −{{sfrac|3|128}} = −{{val|0.02343750}}…
|-
| 6 || 7 || 5 || 10 || {{sfrac|{{val|17649}}|{{val|100000}}}} = {{0|−}}{{val|0.17649000}}…
|-
| 13 || 7 || 11 || 10 || −{{sfrac|{{val|3110989593}}|{{val|100000000000}}}} = −{{val|0.03110990}}…
|-
| 1 || 11 || 1 || 10 || {{sfrac|1|10}} = {{0|−}}{{val|0.10000000}}…
|-
| 1 || 13 || 1 || 10 || {{sfrac|3|10}} = {{0|−}}{{val|0.30000000}}…
|-
| 8 || 13 || 9 || 10 || −{{sfrac|{{val|184269279}}|{{val|1000000000}}}} = −{{val|0.18426928}}…
|-
| 9 || 13 || 10 || 10 || {{sfrac|{{val|604499373}}|{{val|10000000000}}}} = {{0|−}}{{val|0.06044994}}…
|-
| 1 || 17 || 1 || 10 || {{sfrac|7|10}} = {{0|−}}{{val|0.70000000}}…
|-
| 4 || 17 || 5 || 10 || −{{sfrac|{{val|16479}}|{{val|100000}}}} = −{{val|0.16479000}}…
|-
| 9 || 17 || 11 || 10 || {{sfrac|{{val|18587876497}}|{{val|100000000000}}}} = {{0|−}}{{val|0.18587876}}…
|-
| 3 || 19 || 4 || 10 || −{{sfrac|3141|{{val|10000}}}} = −{{val|0.31410000}}…
|-
| 4 || 19 || 5 || 10 || {{sfrac|{{val|30321}}|{{val|100000}}}} = {{0|−}}{{val|0.30321000}}…
|-
| 7 || 19 || 9 || 10 || −{{sfrac|{{val|106128261}}|{{val|1000000000}}}} = −{{val|0.10612826}}…
|-
| 2 || 23 || 3 || 10 || −{{sfrac|471|1000}} = −{{val|0.47100000}}…
|-
| 3 || 23 || 4 || 10 || {{sfrac|2167|{{val|10000}}}} = {{0|−}}{{val|0.21670000}}…
|-
| 2 || 29 || 3 || 10 || −{{sfrac|159|1000}} = −{{val|0.15900000}}…
|-
| 2 || 31 || 3 || 10 || −{{sfrac|39|1000}} = −{{val|0.03900000}}…
|}

== Known digits ==
This is a table of recent records in calculating digits of <math>\ln(2)</math>. As of December 2018, it has been calculated to more digits than any other natural logarithm
<ref>{{cite web|url=http://www.numberworld.org/y-cruncher/|title=y-cruncher|author=|date=|work=numberworld.org|accessdate=10 December 2018}}</ref> <ref>{{cite web|url=http://www.numberworld.org/digits/Log(2)/|title=Natural log of 2|author=|date=|work=numberworld.org|accessdate=10 December 2018}}</ref> of a natural number, except that of 1.

{| class="wikitable"
|-
! 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
|}
==See also==
*[[Rule of 72#Continuous compounding]], in which ln(2) figures prominently

*[[Half-life#Formulas for half-life in exponential decay]], in which ln(2) figures prominently

*[[Erdős–Moser equation]]: all solutions must come from a [[convergent (continued fraction)|convergent]] of ln(2).

==References==
* {{cite journal
|first1=Richard P.
|last1=Brent
|title=Fast multiple-precision evaluation of elementary functions
|journal=J. ACM
|volume=23
|issue=2
|year=1976
|pages=242–251
|doi=10.1145/321941.321944
|mr=0395314
}}
* {{cite journal
|first1=Horace S.
|last1=Uhler
|title=Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17
|journal=Proc. Natl. Acad. Sci. U.S.A.
|volume=26
|issue=3
|year=1940
|pages=205–212
|mr=0001523
|doi=10.1073/pnas.26.3.205
|pmc=1078033
|pmid=16588339
}}
* {{cite journal
|first1=Dura W.
|last1=Sweeney
|title=On the computation of Euler's constant
|journal=Mathematics of Computation
|year=1963
|volume=17
|issue=82
|pages=170–178
|doi=10.1090/S0025-5718-1963-0160308-X 
|mr=0160308
}}
* {{cite journal
|first1=Marc
|last1=Chamberland
|title=Binary BBP-formulae for logarithms and generalized Gaussian–Mersenne primes
|journal=Journal of Integer Sequences
|url=http://www.emis.ams.org/journals/JIS/VOL6/Chamberland/chamberland60.pdf
|volume=6
|page=03.3.7
|year=2003
|mr=2046407
}}
* {{cite journal
|first1=Boris
|last1=Gourévitch
|first2=Jesús
|last2=Guillera Goyanes
|title=Construction of binomial sums for {{pi}} and polylogarithmic constants inspired by BBP formulas
|journal=Applied Math. E-Notes
|volume=7
|year=2007
|url=http://www.math.nthu.edu.tw/~amen/2007/061028-2.pdf
|mr=2346048
|pages=237–246}}
* {{cite journal
|first1=Qiang
|last1=Wu
|title=On the linear independence measure of logarithms of rational numbers
|journal=Mathematics of Computation
|volume=72
|issue=242
|pages=901–911
|doi=10.1090/S0025-5718-02-01442-4
|year=2003
}}
{{reflist}}

==External links==
* {{MathWorld|urlname=NaturalLogarithmof2|title=Natural logarithm of 2}}
* {{planetmath reference|id=10699|title=table of natural logarithms}}
* {{cite web|url=http://numbers.computation.free.fr/Constants/Log2/log2.html|title=The logarithm constant:log 2|last1=Gourdon|first1=Xavier|last2=Sebah|first2=Pascal}}

{{Irrational number}}

{{DEFAULTSORT:Natural Logarithm Of 2}}
[[Category:Logarithms]]
[[Category:Mathematical constants]]
[[Category:Real transcendental numbers]]