2的自然对数

ln2（OEIS數列A002162）约为：

2的自然对数

2的自然对数
數表—无理数 ${\displaystyle \color {blue}{\sqrt {2}}}$ - ${\displaystyle \color {blue}\varphi }$ - ${\displaystyle \color {blue}{\sqrt {3}}}$ - ${\displaystyle \color {blue}{\sqrt {5}}}$ - ${\displaystyle \color {blue}\delta _{S}}$ - ${\displaystyle \color {blue}e}$ - ${\displaystyle \color {blue}\pi }$
識別
種類	無理數
符號	${\displaystyle \ln {2}}$
性質
連分數	[0; 1, 2, 3, 1, 6, 3, 1, 1, 2, 1, 1, 1, 1, 3, 10] （OEIS數列A016730） ${\displaystyle 0+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{6+\ddots }}}}}}}}}}}$
以此為根的多項式或函數	${\displaystyle e^{x}-2=0}$^[1]
表示方式
值	${\displaystyle \ln {2}\approx }$0.693147180...
二进制	0.101100010111001000010111…
十进制	0.693147180559945309417232…
十六进制	0.B17217F7D1CF79ABC9E3B398…
查 论 编

${\displaystyle \ln 2\approx 0.693147}$

使用对数公式

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

可以求出log2，它约为：（OEIS數列A007524）

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

數學家理查德·施羅培爾（英语：Richard Schroeppel）在1972年證明，不尋常數的自然密度等於 ${\displaystyle \ln 2}$。換言之，若 ${\displaystyle u(n)}$ 表示不大於 ${\displaystyle n}$ 的自然數之中，有多少個數 ${\displaystyle a}$ 具有大於 ${\displaystyle {\sqrt {a}}}$ 的質因數，則有：

${\displaystyle \lim _{n\rightarrow \infty }{\frac {u(n)}{n}}=\ln(2)=0.693147\dots \,.}$

公式

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

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

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

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

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

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

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

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

${\displaystyle \gamma }$ 是欧拉-马歇罗尼常数，${\displaystyle \zeta }$ 是黎曼ζ函數。

${\displaystyle \ln 2=\sum _{k\geq 1}{\frac {1}{k2^{k}}}.}$ ^[2]:31

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

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

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

积分公式

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle \gamma }$ 是欧拉-马歇罗尼常数。

其他公式

用皮尔斯展开式（A091846）表达ln2：

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

用恩格尔展开式A059180表达ln2：

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

用余切展开式A081785表达ln2：

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

其他對數

範例

10的自然對數

參考文獻

1. Wolfram, Stephen. "e^x-2=0". from Wolfram Alpha: Computational Knowledge Engine, Wolfram Research （英语）. 
2. Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. §2.2 Integer Relation Detection. Experimental Mathematics in Action. A K Peters/CRC Press. 2007: pp. 29-31. ISBN 978-1568812717.  引文格式1维护：冗余文本 (link)

• 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. (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 [2011-01-08]. MR2346048. （原始内容 (PDF)于2020-02-06）. 
• 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. 
• table of natural logarithms. PlanetMath. 
• Gourdon, Xavier; Sebah, Pascal. The logarithm constant:log 2. [2011-01-08]. （原始内容于2020-02-23）. 

參見

• 對數
• 自然對數
• 常用對數
• 超越数