2的自然对数

ln2（A002162）约为：

$\ln 2\approx 0.693147$

使用对数公式

$\log _{b}2={\frac {\ln 2}{\ln b}}.$

可以求出log2，它约为：（A007524）

$\log _{10}2\approx 0.301029995663981195$。

log210约为：

$\log _{2}10={\frac {1}{\log _{10}2}}\approx 3.321928095$ *A020862）。

n	ln n	OEIS
2	0.693147180559945309417232121458	A002162
3	1.09861228866810969139524523692	A002391
4	1.38629436111989061883446424292	A016627
5	1.60943791243410037460075933323	A016628
6	1.79175946922805500081247735838	A016629
7	1.94591014905531330510535274344	A016630
8	2.07944154167983592825169636437	A016631
9	2.19722457733621938279049047384	A016632
10	2.30258509299404568401799145468	A002392

公式

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

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

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

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

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

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

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

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

$\gamma$是欧拉-马歇罗尼常数，$\zeta$是黎曼ζ函數。

$\ln 2=\sum _{k\geq 1}{\frac {1}{k2^{k}}}.$

$\ln 2=\sum _{k\geq 1}\left({\frac {1}{3^{k}}}+{\frac {1}{4^{k}}}\right){\frac {1}{k}}.$

$\ln 2={\frac {2}{3}}+\sum _{k\geq 1}\left({\frac {1}{2k}}+{\frac {1}{4k+1}}+{\frac {1}{8k+4}}+{\frac {1}{16k+12}}\right){\frac {1}{16^{k}}}.$（贝利－波尔温－普劳夫公式）

$\ln 2={\frac {2}{3}}\sum _{k\geq 0}{\frac {1}{(2k+1)9^{k}}}.$（基於反雙曲函數，可參見計算自然對數的級數。）

积分公式

$\int _{0}^{1}{\frac {dx}{1+x}}=\ln 2$

$\int _{1}^{\infty }{\frac {dx}{(1+x^{2})(1+x)^{2}}}={\frac {1}{4}}(1-\ln 2)$

$\int _{0}^{\infty }{\frac {dx}{1+e^{nx}}}={\frac {1}{n}}\ln 2;\int _{0}^{\infty }{\frac {dx}{3+e^{nx}}}={\frac {2}{3n}}\ln 2$

$\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {2}{e^{2x}-1}}\right)=\ln 2$

$\int _{0}^{\infty }e^{-x}{\frac {1-e^{-x}}{x}}dx=\ln 2$

$\int _{0}^{1}\ln {\frac {x^{2}-1}{x\ln x}}dx=-1+\ln 2+\gamma$

$\int _{0}^{\frac {\pi }{3}}\tan xdx=2\int _{0}^{\frac {\pi }{4}}\tan xdx=\ln 2$

$\int _{-{\frac {\pi }{4}}}^{\frac {\pi }{4}}\ln(\sin x+\cos x)dx=-{\frac {\pi }{4}}\ln 2$

$\int _{0}^{1}x^{2}\ln(1+x)dx={\frac {2}{3}}\ln 2-{\frac {5}{18}}$

$\int _{0}^{1}x\ln(1+x)\ln(1-x)dx={\frac {1}{4}}-\ln 2$

$\int _{0}^{1}x^{3}\ln(1+x)\ln(1-x)dx={\frac {13}{96}}-{\frac {2}{3}}\ln 2$

$\int _{0}^{1}{\frac {\ln x}{(1+x)^{2}}}dx=-\ln 2$

$\int _{0}^{1}{\frac {\ln(1+x)-x}{x^{2}}}dx=1-2\ln 2$

$\int _{0}^{1}{\frac {dx}{x(1-\ln x)(1-2\ln x)}}=\ln 2$

$\int _{1}^{\infty }{\frac {\ln \ln x}{x^{3}}}dx=-{\frac {1}{2}}(\gamma +\ln 2)$

$\gamma$是欧拉-马歇罗尼常数。

其他公式

用皮尔斯展开式（A091846）表达ln2：

$\log 2={\frac {1}{1}}-{\frac {1}{1\cdot 3}}+{\frac {1}{1\cdot 3\cdot 12}}-\ldots$.

用恩格尔展开式A059180表达ln2：

$\log 2={\frac {1}{2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{2\cdot 3\cdot 7}}+{\frac {1}{2\cdot 3\cdot 7\cdot 9}}+\ldots$.

用余切展开式A081785表达ln2：

$\log 2=\cot(\operatorname {arccot} 0-\operatorname {arccot} 1+\operatorname {arccot} 5-\operatorname {arccot} 55+\operatorname {arccot} 14187-\ldots )$.

其他對數

參考文獻

• Brent, Richard P. Fast multiple-precision evaluation of elementary functions. J. ACM. 1976, 23 (2): 242–251. doi:10.1145/321941.321944. MR0395314. 
• Uhler, Horace S. Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17. Proc. Nat. Acac. Sci. U. S. A. 1940, 26: 205–212. MR0001523. 
• Sweeney, Dura W. On the computation of Euler's constant. Mathematics of Computation. 1963, 17. MR0160308. 
• Chamberland, Marc. Binary BBP-formulae for logarithms and generalized Gaussian-Mersenne primes (PDF). Journal of Integer Sequences. 2003, 6: 03.3.7 [2011-01-08]. MR2046407. （原始内容 (PDF)存档于2011-06-06）. 
• Gourévitch, Boris; Guillera Goyanes, Jesus. Construction of binomial sums for π and polylogarithmic constants inspired by BBP formulas (PDF). Applied Math. E-Notes. 2007, 7: 237–246. MR2346048. 
• Wu, Qiang. On the linear independence measure of logarithms of rational numbers. Mathematics of Computation. 2003, 72 (242): 901–911. doi:10.1090/S0025-5718-02-01442-4. 

外部連結

• 埃里克·韦斯坦因. Natural logarithm of 2. MathWorld. 
• PlanetMath上table of natural logarithms的資料。

• Gourdon, Xavier; Sebah, Pascal. The logarithm constant:log 2. 

參見

• 對數
• 自然對數
• 常用對數
• 超越数