Natural logarithm of 2

:$\backslash ln\; 2\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^\{n+1\}\}\{n\}=1-\backslash frac12+\backslash frac13-\backslash frac14+\backslash frac15-\backslash frac16+\backslash cdots.$ This is the well-known "alternating harmonic series".

:$\backslash ln\; 2\; =\; \backslash frac\{1\}\{2\}\; +\backslash frac\{1\}\{2\}\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^\{n+1\}\}\{n(n+1)\}.$

:$\backslash ln\; 2\; =\; \backslash frac\{5\}\{8\}\; +\backslash frac\{1\}\{2\}\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^\{n+1\}\}\{n(n+1)(n+2)\}.$

:$\backslash ln\; 2\; =\; \backslash frac\{2\}\{3\}\; +\backslash frac\{3\}\{4\}\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^\{n+1\}\}\{n(n+1)(n+2)(n+3)\}.$

:$\backslash ln\; 2\; =\; \backslash frac\{131\}\{192\}\; +\backslash frac\{3\}\{2\}\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^\{n+1\}\}\{n(n+1)(n+2)(n+3)(n+4)\}.$

:$\backslash ln\; 2\; =\; \backslash frac\{661\}\{960\}\; +\backslash frac\{15\}\{4\}\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^\{n+1\}\}\{n(n+1)(n+2)(n+3)(n+4)(n+5)\}.$

:$\backslash ln\; 2\; =\; \backslash frac\{2\}\{3\}\backslash left(1+\backslash frac\{2\}\{4^3-4\}+\backslash frac\{2\}\{8^3-8\}+\backslash frac\{2\}\{12^3-12\}+\backslash dots\backslash right)\; .$

:$\backslash ln\; 2\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{2^\{n\}n\}.$

:$\backslash ln\; 2\; =\; 1\; -\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{2^\{n\}n(n+1)\}.$

:$\backslash ln\; 2\; =\; \backslash frac\{1\}\{2\}\; +\; 2\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{2^\{n\}n(n+1)(n+2)\}\; .$

:$\backslash ln\; 2\; =\; \backslash frac\{5\}\{6\}\; -\; 6\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{2^\{n\}n(n+1)(n+2)(n+3)\}\; .$

:$\backslash ln\; 2\; =\; \backslash frac\{7\}\{12\}\; +\; 24\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{2^\{n\}n(n+1)(n+2)(n+3)(n+4)\}\; .$

:$\backslash ln\; 2\; =\; \backslash frac\{47\}\{60\}\; -\; 120\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{2^\{n\}n(n+1)(n+2)(n+3)(n+4)(n+5)\}\; .$

:$\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{1\}\{(2n+1)(2n+2)\}\; =\; \backslash ln\; 2.$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{n(4n^2-1)\}\; =\; 2\backslash ln\; 2\; -1.$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^n\}\{n(4n^2-1)\}\; =\; \backslash ln\; 2\; -1.$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^n\}\{n(9n^2-1)\}\; =\; 2\backslash ln\; 2\; -\backslash frac\{3\}\{2\}.$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{4n^2-2n\}\; =\; \backslash ln\; 2.$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{2(-1)^\{n+1\}(2n-1)+1\}\{8n^2-4n\}\; =\; \backslash ln\; 2.$

:$\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-1)^\{n\}\}\{3n+1\}\; =\; \backslash frac\{\backslash ln\; 2\}\{3\}+\backslash frac\{\backslash pi\}\{3\backslash sqrt\{3\}\}.$

:$\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-1)^\{n\}\}\{3n+2\}\; =\; -\backslash frac\{\backslash ln\; 2\}\{3\}+\backslash frac\{\backslash pi\}\{3\backslash sqrt\{3\}\}.$

:$\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-1)^\{n\}\}\{(3n+1)(3n+2)\}\; =\; \backslash frac\{2\backslash ln\; 2\}\{3\}.$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{\backslash sum\_\{k=1\}^n\; k^2\}\; =\; 18\; -\; 24\; \backslash ln\; 2$ using $\backslash lim\_\{N\backslash rightarrow\; \backslash infty\}\; \backslash sum\_\{n=N\}^\{2N\}\; \backslash frac\{1\}\{n\}\; =\; \backslash ln\; 2$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{4n^2-3n\}\; =\; \backslash ln\; 2\; +\; \backslash frac\{\backslash pi\}\{6\}$ (sums of the reciprocals of decagonal numbers)

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{n\}[\backslash zeta(2n)-1]\; =\; \backslash ln\; 2.$

:$\backslash sum\_\{n=2\}^\backslash infty\; \backslash frac\{1\}\{2^n\}[\backslash zeta(n)-1]\; =\; \backslash ln\; 2\; -\backslash frac\{1\}\{2\}.$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{2n+1\}[\backslash zeta(2n+1)-1]\; =\; 1-\backslash gamma-\backslash frac\{\backslash ln\; 2\}\{2\}.$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{2^\{2n-1\}(2n+1)\}\backslash zeta(2n)\; =\; 1-\backslash ln\; 2.$

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

:$\backslash ln\; 2\; =\; \backslash frac\{2\}\{3\}\; +\; \backslash frac\{1\}\{2\}\; \backslash sum\_\{k\; =\; 1\}^\backslash infty\; \backslash left(\backslash frac\{1\}\{2k\}+\backslash frac\{1\}\{4k+1\}+\backslash frac\{1\}\{8k+4\}+\backslash frac\{1\}\{16k+12\}\backslash right)\; \backslash frac\{1\}\{16^k\}\; .$

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

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

:$\backslash ln\; 2\; =\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{(-1)^\{n-1\}\}\{n\}.$

:$\backslash ln\; 2\; =\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{2^\{n\}n\}.$

:$\backslash ln\; 2\; =\; \backslash frac\{2\}\{3\}\; \backslash sum\_\{k\; =\; 0\}^\backslash infty\; \backslash frac\{1\}\{9^\{k\}(2k+1)\}.$

Applying them to $\backslash textstyle\; 2\; =\; \backslash frac\{3\}\{2\}\; \backslash cdot\; \backslash frac\{4\}\{3\}$ gives:

:$\backslash ln\; 2\; =\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{(-1)^\{n-1\}\}\{2^n\; n\}\; +\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{(-1)^\{n-1\}\}\{3^n\; n\}\; .$

:$\backslash ln\; 2\; =\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{3^n\; n\}\; +\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{4^n\; n\}\; .$

:$\backslash ln\; 2\; =\; \backslash frac\{2\}\{5\}\; \backslash sum\_\{k\; =\; 0\}^\backslash infty\; \backslash frac\{1\}\{25^\{k\}(2k+1)\}\; +\; \backslash frac\{2\}\{7\}\; \backslash sum\_\{k\; =\; 0\}^\backslash infty\; \backslash frac\{1\}\{49^\{k\}(2k+1)\}\; .$

Applying them to $\backslash textstyle\; 2\; =\; (\backslash sqrt\{2\})^2$ gives:

:$\backslash ln\; 2\; =\; 2\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{(-1)^\{n-1\}\}\{(\backslash sqrt\{2\}\; +\; 1)^n\; n\}\; .$

:$\backslash ln\; 2\; =\; 2\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{(2\; +\; \backslash sqrt\{2\})^n\; n\}\; .$

:$\backslash ln\; 2\; =\; \backslash frac\{4\}\{3\; +\; 2\; \backslash sqrt\{2\}\}\; \backslash sum\_\{k\; =\; 0\}^\backslash infty\; \backslash frac\{1\}\{(17\; +\; 12\; \backslash sqrt\{2\})^\{k\}(2k+1)\}\; .$

Applying them to $\backslash textstyle\; 2\; =\; \{\; \backslash left(\; \backslash frac\{16\}\{15\}\; \backslash right)\; \}^\{7\}\; \backslash cdot\; \{\; \backslash left(\; \backslash frac\{81\}\{80\}\; \backslash right)\; \}^\{3\}\; \backslash cdot\; \{\; \backslash left(\; \backslash frac\{25\}\{24\}\; \backslash right)\; \}^\{5\}$ gives:

:$\backslash ln\; 2\; =\; 7\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{(-1)^\{n-1\}\}\{15^n\; n\}\; +\; 3\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{(-1)^\{n-1\}\}\{80^n\; n\}\; +\; 5\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{(-1)^\{n-1\}\}\{24^n\; n\}\; .$

:$\backslash ln\; 2\; =\; 7\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{16^n\; n\}\; +\; 3\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{81^n\; n\}\; +\; 5\; \backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{25^n\; n\}\; .$

:$\backslash ln\; 2\; =\; \backslash frac\{14\}\{31\}\; \backslash sum\_\{k\; =\; 0\}^\backslash infty\; \backslash frac\{1\}\{961^\{k\}(2k+1)\}\; +\; \backslash frac\{6\}\{161\}\; \backslash sum\_\{k\; =\; 0\}^\backslash infty\; \backslash frac\{1\}\{25921^\{k\}(2k+1)\}\; +\; \backslash frac\{10\}\{49\}\; \backslash sum\_\{k\; =\; 0\}^\backslash infty\; \backslash frac\{1\}\{2401^\{k\}(2k+1)\}\; .$

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

:$\backslash int\_0^1\; \backslash frac\{dx\}\{1+x\}\; =\; \backslash int\_1^2\; \backslash frac\{dx\}\{x\}\; =\; \backslash ln\; 2$

:$\backslash int\_0^\backslash infty\; e^\{-x\}\backslash frac\{1-e^\{-x\}\}\{x\}\; \backslash ,\; dx=\; \backslash ln\; 2$

:$\backslash int\_0^\backslash infty\; 2^\{-x\}\; dx=\; \backslash frac\{1\}\{\backslash ln\; 2\}$

:$\backslash int\_0^\backslash frac\{\backslash pi\}\{3\}\; \backslash tan\; x\; \backslash ,\; dx=2\backslash int\_0^\backslash frac\{\backslash pi\}\{4\}\; \backslash tan\; x\; \backslash ,\; dx\; =\; \backslash ln\; 2$

:$-\backslash frac\{1\}\{\backslash pi\; i\}\backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash ln\; x\; \backslash ln\backslash ln\; x\}\{(x+1)^2\}\; \backslash ,\; dx=\; \backslash ln\; 2$

The Pierce expansion is {{OEIS2C|A091846}}

:$\backslash ln\; 2\; =\; 1\; -\backslash frac\{1\}\{1\backslash cdot\; 3\}+\backslash frac\{1\}\{1\backslash cdot\; 3\backslash cdot\; 12\}\; -\backslash cdots.$

The Engel expansion is {{OEIS2C|A059180}}

:$\backslash ln\; 2\; =\; \backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{2\backslash cdot\; 3\}\; +\; \backslash frac\{1\}\{2\backslash cdot\; 3\backslash cdot\; 7\}\; +\; \backslash frac\{1\}\{2\backslash cdot\; 3\backslash cdot\; 7\backslash cdot\; 9\}+\backslash cdots.$

The cotangent expansion is {{OEIS2C|A081785}}

:$\backslash ln\; 2\; =\; \backslash cot(\{\backslash arccot(0)\; -\backslash arccot(1)\; +\; \backslash arccot(5)\; -\; \backslash arccot(55)\; +\; \backslash arccot(14187)\; -\backslash cdots\}).$

The simple continued fraction expansion is {{OEIS2C|A016730}}

:$\backslash ln\; 2\; =\; \backslash left[\; 0;\; 1,\; 2,\; 3,\; 1,\; 6,\; 3,\; 1,\; 1,\; 2,\; 1,\; 1,\; 1,\; 1,\; 3,\; 10,\; 1,\; 1,\; 1,\; 2,\; 1,\; 1,\; 1,\; 1,\; 3,\; 2,\; 3,\; 1,...\backslash right]$,

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:

:$\backslash ln\; 2\; =\; \backslash left[\; 0;1,2,3,1,5,\backslash tfrac\{2\}\{3\},7,\backslash tfrac\{1\}\{2\},9,\backslash tfrac\{2\}\{5\},...,2k-1,\backslash frac\{2\}\{k\},...\backslash right]$,{{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|s2cid=17758274}}

:also expressible as

:$\backslash ln\; 2\; =\; \backslash cfrac\{1\}\; \{1+\backslash cfrac\{1\}\; \{2+\backslash cfrac\{1\}\; \{3+\backslash cfrac\{2\}\; \{2+\backslash cfrac\{2\}\; \{5+\backslash cfrac\{3\}\; \{2+\backslash cfrac\{3\}\; \{7+\backslash cfrac\{4\}\; \{2+\backslash ddots\}\}\}\}\}\}\}\}$

= \cfrac{2} {3-\cfrac{1^2} {9-\cfrac{2^2} {15-\cfrac{3^2} {21-\ddots}}}}

Given a value of {{math|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 {{math|c}} based on their factorizations

:$c=2^i3^j5^k7^l\backslash cdots\backslash rightarrow\; \backslash ln(c)=i\backslash ln(2)+j\backslash ln(3)+k\backslash ln(5)+l\backslash ln(7)+\backslash cdots$

This employs

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 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 conversions.

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

:$s\backslash ln\; p\; -t\backslash ln\; q\; =\; \backslash ln\backslash left(1+\backslash frac\{d\}\{q^t\}\backslash right)\; =\; \backslash sum\_\{m=1\}^\backslash infty\; \backslash frac\{(-1)^\{m+1\}\}\{m\}\backslash left(\backslash frac\{d\}\{q^t\}\backslash right)^m\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{2\}\{2n+1\}\; \{\backslash left(\backslash frac\{d\}\{2\; q^t\; +\; d\}\backslash right)\}^\{2n+1\}\; .$

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

:$2\backslash ln\; 3\; =\; 3\backslash ln\; 2\; -\backslash sum\_\{k\backslash ge\; 1\}\backslash frac\{(-1)^k\}\{8^\{k\}k\}\; =\; 3\backslash ln\; 2\; +\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{2\}\{2n+1\}\; \{\backslash left(\backslash frac\{1\}\{2\; \backslash cdot\; 8\; +\; 1\}\backslash right)\}^\{2n+1\}\; .$

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

Starting from the natural logarithm of {{math|q {{=}} 10}} one might use these parameters:

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

• Rule of 72#Continuous compounding, in which {{math|ln 2}} figures prominently
• Half-life#Formulas for half-life in exponential decay, in which {{math|ln 2}} figures prominently
• Erdős–Moser equation: all solutions must come from a convergent of {{math|ln 2}}.

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{{reflist}}

• {{MathWorld|urlname=NaturalLogarithmof2|title=Natural logarithm of 2}}
• {{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