Natural Logarithm

The Logarithm having base e, where

$\begin{displaymath} e=2.718281828\ldots, \end{displaymath}$

(1)

which can be defined

$\begin{displaymath} \ln x \equiv \int_1^x {dt\over t} \end{displaymath}$

(2)

for

. The natural logarithm can also be defined for Complex Numbers as

$\begin{displaymath} \ln z \equiv \ln\vert z\vert+i \arg(z), \end{displaymath}$

(3)

where $\vert z\vert$ is the Modulus and $\arg(z)$ is the Argument. The natural logarithm is especially useful in Calculus because its Derivative is given by the simple equation

$\begin{displaymath} {d\over dx}\ln x={1\over x}, \end{displaymath}$

(4)

whereas logarithms in other bases have the more complicated Derivative

$\begin{displaymath} {d\over dx}\log_b x={1\over x\ln b}. \end{displaymath}$

(5)

The Mercator Series

$\begin{displaymath} \ln(1+x)=x-{\textstyle{1\over 2}}x^2+{\textstyle{1\over 3}} x^3-\ldots \end{displaymath}$

(6)

gives a Taylor Series for the natural logarithm.

Continued Fraction representations of logarithmic functions include

$\begin{displaymath} \ln(1+x) = {x\over 1+{\strut\displaystyle 1^2x\over\strut\di... ...trut\displaystyle 3^2x\over\strut\displaystyle 7+\ldots}}}}}}} \end{displaymath}$

(7)

$\begin{displaymath} \ln\left({1+x\over 1-x}\right)= {2x\over \strut\displaystyle... ...trut\displaystyle 16x^2\over\strut\displaystyle 9-\ldots}}}}}. \end{displaymath}$

(8)

For a Complex Number , the natural logarithm satisfies

$\begin{displaymath} \ln z = \ln[re^{i(\theta +2n\pi)}] = \ln r+i(\theta +2n\pi) \end{displaymath}$

(9)

$\begin{displaymath} PV(\ln z) = \ln r+i\theta, \end{displaymath}$

(10)

where

is the Principal Value.

Some special values of the natural logarithm are

$\begin{displaymath} \ln 1 =0 \end{displaymath}$

(11)

$\begin{displaymath} \ln 0 =-\infty \end{displaymath}$

(12)

$\begin{displaymath} \ln(-1)=\pi i \end{displaymath}$

(13)

$\begin{displaymath} \ln(\pm i)=\pm{\textstyle{1\over 2}}\pi i. \end{displaymath}$

(14)

An identity for the natural logarithm of 2 discovered using the PSLQ Algorithm is

$\begin{displaymath} (\ln 2)^2=2\sum_{i=1}^\infty {p_i\over 2^i i^2} \quad \{p_i\}=\{\,\overline{2, -10, -7, -10, 2, -1}\,\}, \end{displaymath}$

(15)

where $\{p_i\}$ is given by the periodic sequence obtained by appending copies of $\{2, -10, -7, -10, 2, -1\}$ (in other words, $p_i\equiv p_{[(i-1) {\rm\ (mod\ 6})]+1}$ for

) (Bailey et al. 1995, Bailey and Plouffe).

See also e, Jensen's Formula, Lg, Logarithm

References

Bailey, D.; Borwein, P.; and Plouffe, S. ``On the Rapid Computation of Various Polylogarithmic Constants.'' http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps.

Bailey, D. and Plouffe, S. ``Recognizing Numerical Constants.'' http://www.cecm.sfu.ca/organics/papers/bailey/.