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| Line 1: |
Line 1: |
| The decimal value of the [[natural logarithm]] of [[2 (number)|2]] {{OEIS|A002162}}
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| is approximately
| |
| :<math>\ln 2 \approx 0.69314718055...</math>
| |
| as shown in the first line of the table below.
| |
| The logarithm 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 0.301029995663981195.</math>
| |
| The inverse of this number is the [[binary logarithm]] of 10:
| |
| :<math> \log_2 10=1/\log_{10} 2 \approx 3.321928095</math> ({{OEIS2C|A020862}}).
| |
| {{Table
| |
| |type=class="wikitable"
| |
| |hdrs=number!!approximate natural logarithm!![[OEIS]]
| |
| |row1=[[2 (number)|2]] {{!!}} 0.693147180559945309417232121458 {{!!}} {{OEIS link|A002162}}
| |
| |row2=[[3 (number)|3]]{{!!}}1.09861228866810969139524523692{{!!}} {{OEIS link|A002391}}
| |
| |row3=[[4 (number)|4]]{{!!}}1.38629436111989061883446424292{{!!}} {{OEIS link|A016627}}
| |
| |row4=[[5 (number)|5]]{{!!}}1.60943791243410037460075933323{{!!}} {{OEIS link|A016628}}
| |
| |row5=[[6 (number)|6]]{{!!}}1.79175946922805500081247735838{{!!}} {{OEIS link|A016629}}
| |
| |row6=[[7 (number)|7]]{{!!}}1.94591014905531330510535274344{{!!}} {{OEIS link|A016630}}
| |
| |row7=[[8 (number)|8]]{{!!}}2.07944154167983592825169636437{{!!}} {{OEIS link|A016631}}
| |
| |row8=[[9 (number)|9]]{{!!}}2.19722457733621938279049047384{{!!}} {{OEIS link|A016632}}
| |
| |row9=[[10 (number)|10]]{{!!}}2.30258509299404568401799145468{{!!}} {{OEIS link|A002392}}
| |
| }}
| |
| | |
| ==Series representations==
| |
| | |
| :<math>\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = \sum_{n=0}^\infty \frac{1}{(2n+1)(2n+2)} = \ln 2.</math> | |
| | |
| :<math>\sum_{n=1}^\infty \frac{(-1)^n}{(n+1)(n+2)} = 2\ln 2 -1.</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 -\frac{3}{2}.</math>
| |
| | |
| :<math>\sum_{n=2}^\infty \frac{1}{2^n}[\zeta(n)-1] = \ln 2 -\frac{1}{2}.</math>
| |
| | |
| :<math>\sum_{n=1}^\infty \frac{1}{2n+1}[\zeta(n)-1] = 1-\gamma-\frac{1}{2}\ln 2.</math>
| |
| | |
| :<math>\sum_{n=1}^\infty \frac{1}{2^{2n}(2n+1)}\zeta(2n) = \frac{1}{2}(1-\ln 2).</math>
| |
| | |
| :<math>\ln 2 = \sum_{k\ge 1} \frac{1}{k2^k}.</math>
| |
| | |
| :<math>\ln 2 = \sum_{k\ge 1}\left(\frac{1}{3^k}+\frac{1}{4^k}\right)\frac{1}{k}.</math>
| |
| | |
| :<math>\ln 2 = \frac{2}{3}+\sum_{k\ge 1}\left(\frac{1}{2k}+\frac{1}{4k+1}+\frac{1}{8k+4}+\frac{1}{16k+12}\right)\frac{1}{16^k}.</math>
| |
| | |
| :<math>\ln 2 = \frac{2}{3} \sum_{k\ge 0} \frac{1}{(2k+1)9^k}.</math>
| |
| | |
| (<math>\gamma</math> is the [[Euler constant|Euler–Mascheroni constant]]
| |
| and <math>\zeta</math> [[Riemann zeta function|Riemann's zeta function]]).
| |
| | |
| Some [[Bailey–Borwein–Plouffe formula|Bailey–Borwein–Plouffe (BBP)-type representations]] fall also into this category.
| |
| | |
| ==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}{4}(1-\ln 2).</math>
| |
| | |
| :<math>\int_0^\infty \frac{dx}{1+e^{nx}} = \frac{1}{n}\ln 2;
| |
| \int_0^\infty \frac{dx}{3+e^{nx}} = \frac{2}{3n}\ln 2.</math>
| |
| | |
| :<math>\int_0^\infty \frac{1}{e^x-1}-\frac{2}{e^{2x}-1}\,dx=\ln 2.</math>
| |
| | |
| :<math>\int_0^\infty e^{-x}\frac{1-e^{-x}}{x} \, dx= \ln 2.</math>
| |
| | |
| :<math>\int_0^1 \ln\frac{x^2-1}{x\ln x}dx=-1+\ln 2+\gamma.</math>
| |
| | |
| :<math>\int_0^{\pi/3} \tan x \, dx=2\int_0^{\pi/4} \tan x \, dx=\ln 2.</math>
| |
| | |
| :<math>\int_{-\pi/4}^{\pi/4} \ln(\sin x+\cos x)\,dx=-\frac{\pi}{4}\ln 2.</math>
| |
| | |
| :<math>\int_0^1 x^2\ln(1+x)\,dx=\frac{2}{3}\ln 2-\frac{5}{18}.</math>
| |
| | |
| :<math>\int_0^1 x\ln(1+x)\ln(1-x)\,dx=\frac{1}{4}-\ln 2.</math>
| |
| | |
| :<math>\int_0^1 x^3\ln(1+x)\ln(1-x)\,dx=\frac{13}{96}-\frac{2}{3}\ln 2.</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\ln2.</math>
| |
| | |
| :<math>\int_0^1 \frac{dx}{x(1-\ln x)(1-2\ln x)} = \ln 2.</math>
| |
| | |
| :<math>\int_1^\infty \frac{\ln\ln x}{x^3}\,dx = -\frac{1}{2}(\gamma+\ln 2).</math>
| |
| | |
| (<math>\gamma</math> is the [[Euler constant|Euler–Mascheroni constant]]).
| |
| | |
| ==Other representations==
| |
| The Pierce expansion is {{OEIS2C|A091846}}
| |
| :<math> \ln 2 = \frac{1}{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 (\operatorname{arccot} 0 -\operatorname{arccot} 1 + \operatorname{arccot} 5 - \operatorname{arccot} 55 + \operatorname{arccot} 14187 -\cdots). </math>
| |
| As an infinite sum of fractions:<ref>"The Penguin's Dictionary of Curious and Interesting Numbers" by David Wells, page 29.</ref>
| |
| :<math> \ln 2 = \frac{1}{1} -\frac{1}{2} +\frac{1}{3} -\frac{1}{4} +\frac{1}{5} -\cdots. </math>
| |
| This [[generalized continued fraction]]:
| |
| :<math> \ln 2 = \left[ 0;1,2,3,1,5,\frac{2}{3},7,\frac{1}{2},9,\frac{2}{5},...,2k-1,\frac{2}{k},...\right] </math>,<ref>[http://www.kurims.kyoto-u.ac.jp/EMIS/journals/EM/expmath/volumes/13/13.3/BorweinCrandallFee.pdf On the Ramanujan AGM Fraction, I: The Real-Parameter Case. Experimental Mathematics, Vol. 13 (2004), No. 3, pages 278,280.]</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</math>, 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</math> based on their [[Table of prime factors|factorizations]]
| |
| :<math>c=2^i3^j5^k7^\ell\cdots\rightarrow \ln c=i\ln 2+j\ln 3+k\ln 5+\ell\ln 7+\cdots</math>
| |
| | |
| Apart from the logarithms of 2, 3, 5 and 7 shown above, this employs
| |
| {{Table
| |
| |type=class="wikitable sortabel"
| |
| |hdrs=prime!!approximate natural logarithm!![[OEIS]]
| |
| |row1=[[11 (number)|11]] {{!!}} 2.39789527279837054406194357797 {{!!}} {{OEIS link|A016634}}
| |
| |row2=[[13 (number)|13]] {{!!}} 2.56494935746153673605348744157 {{!!}} {{OEIS link|A016636}}
| |
| |row3=[[17 (number)|17]] {{!!}} 2.83321334405621608024953461787 {{!!}} {{OEIS link|A016640}}
| |
| |row4=[[19 (number)|19]] {{!!}} 2.94443897916644046000902743189 {{!!}} {{OEIS link|A016642}}
| |
| |row5=[[23 (number)|23]] {{!!}} 3.13549421592914969080675283181 {{!!}} {{OEIS link|A016646}}
| |
| |row6=[[29 (number)|29]] {{!!}} 3.36729582998647402718327203236 {{!!}} {{OEIS link|A016652}}
| |
| |row7=[[31 (number)|31]] {{!!}} 3.43398720448514624592916432454 {{!!}} {{OEIS link|A016654}}
| |
| |row8=[[37 (number)|37]] {{!!}} 3.61091791264422444436809567103 {{!!}} {{OEIS link|A016660}}
| |
| |row9=[[41 (number)|41]] {{!!}} 3.71357206670430780386676337304 {{!!}} {{OEIS link|A016664}}
| |
| |row10=[[43 (number)|43]] {{!!}} 3.76120011569356242347284251335 {{!!}} {{OEIS link|A016666}}
| |
| |row11=[[47 (number)|47]] {{!!}} 3.85014760171005858682095066977 {{!!}} {{OEIS link|A016670}}
| |
| |row12=[[53 (number)|53]] {{!!}} 3.97029191355212183414446913903 {{!!}} {{OEIS link|A016676}}
| |
| |row13=[[59 (number)|59]] {{!!}} 4.07753744390571945061605037372 {{!!}} {{OEIS link|A016682}}
| |
| |row14=[[61 (number)|61]] {{!!}} 4.11087386417331124875138910343 {{!!}} {{OEIS link|A016684}}
| |
| |row15=[[67 (number)|67]] {{!!}} 4.20469261939096605967007199636 {{!!}} {{OEIS link|A016690}}
| |
| |row16=[[71 (number)|71]] {{!!}} 4.26267987704131542132945453251 {{!!}} {{OEIS link|A016694}}
| |
| |row17=[[73 (number)|73]] {{!!}} 4.29045944114839112909210885744 {{!!}} {{OEIS link|A016696}}
| |
| |row18=[[79 (number)|79]] {{!!}} 4.36944785246702149417294554148 {{!!}} {{OEIS link|A016702}}
| |
| |row19=[[83 (number)|83]] {{!!}} 4.41884060779659792347547222329 {{!!}} {{OEIS link|A016706}}
| |
| |row20=[[89 (number)|89]] {{!!}} 4.48863636973213983831781554067 {{!!}} {{OEIS link|A016712}}
| |
| |row21=[[97 (number)|97]] {{!!}} 4.57471097850338282211672162170 {{!!}} {{OEIS link|A016720}}
| |
| }}
| |
| | |
| In a third layer, the logarithms of rational numbers <math>r=a/b</math>
| |
| are computed with <math>\ln r = \ln a -\ln b</math>, and logarithms of roots
| |
| via <math>\ln \sqrt[n]{c} = \frac{1}{n}\ln c</math>.
| |
| | |
| The logarithm of [[2 (number)|2]] is useful in the sense that the powers
| |
| of 2 are rather densely distributed; finding powers <math>2^i</math>
| |
| close to powers <math>b^j</math> of other numbers <math>b</math> is
| |
| comparatively easy, and series representations of <math>\ln b</math>
| |
| are found by coupling <math>2</math> to <math>b</math>
| |
| with [[Logarithmic identities|logarithmic conversions]].
| |
| | |
| ===Example===
| |
| If <math>p^s=q^t+d</math> with some small <math>d</math>,
| |
| then <math>p^s/q^t=1+d/q^t</math> 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{(d/q^t)^m}{m}.
| |
| </math>
| |
| Selecting <math>q=2</math>
| |
| represents <math>\ln p</math> by <math>\ln 2</math> and a series of
| |
| a parameter <math>d/q^t</math> that one wishes to keep small for quick convergence. Taking <math>3^2=2^3+1</math>, for example, generates
| |
| :<math> 2\ln 3 = 3\ln 2-\sum_{k\ge 1}\frac{(-1)^k}{k8^k}.</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}{k8^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}{k16^k}.</math>
| |
| --> | |
| {{Table
| |
| |type=class="wikitable sortable"
| |
| |hdrs=s!!p!!t!!q!!d/q<sup>''t''</sup>
| |
| |row1=1{{!!}}3 {{!!}} 1 {{!!}} 2 {{!!}} 1 / 2 = 0.50000000...
| |
| |row2=1{{!!}}3 {{!!}} 2 {{!!}} 2 {{!!}} −1 / 4 = −0.25000000...
| |
| |row3=2{{!!}}3 {{!!}} 3 {{!!}} 2 {{!!}} 1 / 8 = 0.12500000...
| |
| |row4=5{{!!}}3 {{!!}} 8 {{!!}} 2 {{!!}} −13 / 256 = −0.05078125...
| |
| |row5=12{{!!}}3 {{!!}} 19 {{!!}} 2 {{!!}} 7153 / 524288 = 0.01364326...
| |
| |row6=1{{!!}}5 {{!!}} 2 {{!!}} 2 {{!!}} 1 / 4 = 0.25000000...
| |
| |row7=3{{!!}}5 {{!!}} 7 {{!!}} 2 {{!!}} −3 / 128 = −0.02343750...
| |
| |row8=1{{!!}}7 {{!!}} 2 {{!!}} 2 {{!!}} 3 / 4 = 0.75000000...
| |
| |row9=1{{!!}}7 {{!!}} 3 {{!!}} 2 {{!!}} −1 / 8 = −0.12500000...
| |
| |row10=5{{!!}}7 {{!!}} 14 {{!!}} 2 {{!!}} 423 / 16384 = 0.02581787...
| |
| |row11=1{{!!}}11 {{!!}} 3 {{!!}} 2 {{!!}} 3 / 8 = 0.37500000...
| |
| |row12=2{{!!}}11 {{!!}} 7 {{!!}} 2 {{!!}} −7 / 128 = −0.05468750...
| |
| |row13=11{{!!}}11 {{!!}} 38 {{!!}} 2 {{!!}} 10433763667 / 274877906944 = 0.03795781...
| |
| |row14=1{{!!}}13 {{!!}} 3 {{!!}} 2 {{!!}} 5 / 8 = 0.62500000...
| |
| |row15=1{{!!}}13 {{!!}} 4 {{!!}} 2 {{!!}} −3 / 16 = −0.18750000...
| |
| |row16=3{{!!}}13 {{!!}} 11 {{!!}} 2 {{!!}} 149 / 2048 = 0.07275391...
| |
| |row17=7{{!!}}13 {{!!}} 26 {{!!}} 2 {{!!}} −4360347 / 67108864 = −0.06497423...
| |
| |row18=10{{!!}}13 {{!!}} 37 {{!!}} 2 {{!!}} 419538377 / 137438953472 = 0.00305254...
| |
| |row19=1{{!!}}17 {{!!}} 4 {{!!}} 2 {{!!}} 1 / 16 = 0.06250000...
| |
| |row20=1{{!!}}19 {{!!}} 4 {{!!}} 2 {{!!}} 3 / 16 = 0.18750000...
| |
| |row21=4{{!!}}19 {{!!}} 17 {{!!}} 2 {{!!}} −751 / 131072 = −0.00572968...
| |
| |row22=1{{!!}}23 {{!!}} 4 {{!!}} 2 {{!!}} 7 / 16 = 0.43750000...
| |
| |row23=1{{!!}}23 {{!!}} 5 {{!!}} 2 {{!!}} −9 / 32 = −0.28125000...
| |
| |row24=2{{!!}}23 {{!!}} 9 {{!!}} 2 {{!!}} 17 / 512 = 0.03320312...
| |
| |row25=1{{!!}}29 {{!!}} 4 {{!!}} 2 {{!!}} 13 / 16 = 0.81250000...
| |
| |row26=1{{!!}}29 {{!!}} 5 {{!!}} 2 {{!!}} −3 / 32 = −0.09375000...
| |
| |row27=7{{!!}}29 {{!!}} 34 {{!!}} 2 {{!!}} 70007125 / 17179869184 = 0.00407495...
| |
| |row28=1{{!!}}31 {{!!}} 5 {{!!}} 2 {{!!}} −1 / 32 = −0.03125000...
| |
| |row29=1{{!!}}37 {{!!}} 5 {{!!}} 2 {{!!}} 5 / 32 = 0.15625000...
| |
| |row30=4{{!!}}37 {{!!}} 21 {{!!}} 2 {{!!}} −222991 / 2097152 = −0.10633039...
| |
| |row31=5{{!!}}37 {{!!}} 26 {{!!}} 2 {{!!}} 2235093 / 67108864 = 0.03330548...
| |
| |row32=1{{!!}}41 {{!!}} 5 {{!!}} 2 {{!!}} 9 / 32 = 0.28125000...
| |
| |row33=2{{!!}}41 {{!!}} 11 {{!!}} 2 {{!!}} −367 / 2048 = −0.17919922...
| |
| |row34=3{{!!}}41 {{!!}} 16 {{!!}} 2 {{!!}} 3385 / 65536 = 0.05165100...
| |
| |row35=1{{!!}}43 {{!!}} 5 {{!!}} 2 {{!!}} 11 / 32 = 0.34375000...
| |
| |row36=2{{!!}}43 {{!!}} 11 {{!!}} 2 {{!!}} −199 / 2048 = −0.09716797...
| |
| |row37=5{{!!}}43 {{!!}} 27 {{!!}} 2 {{!!}} 12790715 / 134217728 = 0.09529825...
| |
| |row38=7{{!!}}43 {{!!}} 38 {{!!}} 2 {{!!}} −3059295837 / 274877906944 = −0.01112965...
| |
| | |
| }}
| |
| | |
| Starting from the natural logarithm of <math>q=10</math> one might use these parameters:
| |
| {{Table
| |
| |type=class="wikitable sortable"
| |
| |hdrs=s!!p!!t!!q!!d/q<sup>''t''</sup>
| |
| |row1=10{{!!}}2 {{!!}} 3 {{!!}} 10 {{!!}} 3 / 125 = 0.02400000...
| |
| |row2=21{{!!}}3 {{!!}} 10 {{!!}} 10 {{!!}} 460353203 / 10000000000 = 0.04603532...
| |
| |row3=3{{!!}}5 {{!!}} 2 {{!!}} 10 {{!!}} 1 / 4 = 0.25000000...
| |
| |row4=10{{!!}}5 {{!!}} 7 {{!!}} 10 {{!!}} −3 / 128 = −0.02343750...
| |
| |row5=6{{!!}}7 {{!!}} 5 {{!!}} 10 {{!!}} 17649 / 100000 = 0.17649000...
| |
| |row6=13{{!!}}7 {{!!}} 11 {{!!}} 10 {{!!}} −3110989593 / 100000000000 = −0.03110990...
| |
| |row7=1{{!!}}11 {{!!}} 1 {{!!}} 10 {{!!}} 1 / 10 = 0.10000000...
| |
| |row8=1{{!!}}13 {{!!}} 1 {{!!}} 10 {{!!}} 3 / 10 = 0.30000000...
| |
| |row9=8{{!!}}13 {{!!}} 9 {{!!}} 10 {{!!}} −184269279 / 1000000000 = −0.18426928...
| |
| |row10=9{{!!}}13 {{!!}} 10 {{!!}} 10 {{!!}} 604499373 / 10000000000 = 0.06044994...
| |
| |row11=1{{!!}}17 {{!!}} 1 {{!!}} 10 {{!!}} 7 / 10 = 0.70000000...
| |
| |row12=4{{!!}}17 {{!!}} 5 {{!!}} 10 {{!!}} −16479 / 100000 = −0.16479000...
| |
| |row13=9{{!!}}17 {{!!}} 11 {{!!}} 10 {{!!}} 18587876497 / 100000000000 = 0.18587876...
| |
| |row14=3{{!!}}19 {{!!}} 4 {{!!}} 10 {{!!}} −3141 / 10000 = −0.31410000...
| |
| |row15=4{{!!}}19 {{!!}} 5 {{!!}} 10 {{!!}} 30321 / 100000 = 0.30321000...
| |
| |row16=7{{!!}}19 {{!!}} 9 {{!!}} 10 {{!!}} −106128261 / 1000000000 = −0.10612826...
| |
| |row17=2{{!!}}23 {{!!}} 3 {{!!}} 10 {{!!}} −471 / 1000 = −0.47100000...
| |
| |row18=3{{!!}}23 {{!!}} 4 {{!!}} 10 {{!!}} 2167 / 10000 = 0.21670000...
| |
| |row19=2{{!!}}29 {{!!}} 3 {{!!}} 10 {{!!}} −159 / 1000 = −0.15900000...
| |
| |row20=2{{!!}}31 {{!!}} 3 {{!!}} 10 {{!!}} −39 / 1000 = −0.03900000...
| |
| | |
| }}
| |
| | |
| ==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
| |
| |year=1940
| |
| |pages=205–212
| |
| |url=http://www.pnas.org/content/26/3/205.full.pdf+html
| |
| |mr=0001523
| |
| }}
| |
| * {{cite journal
| |
| |first1=Dura W.
| |
| |last1=Sweeney
| |
| |title=On the computation of Euler's constant
| |
| |journal=Mathematics of Computation
| |
| |year=1963
| |
| |volume=17
| |
| |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=Jesus
| |
| |last2=Guillera Goyanes
| |
| |title=Construction of binomial sums for π 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}}
| |
| | |
| {{DEFAULTSORT:Natural Logarithm Of 2}}
| |
| [[Category:Numbers]]
| |
| [[Category:Logarithms]]
| |
| [[Category:Irrational numbers]]
| |