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Natural Logarithm

The Logarithm having base e, where

\end{displaymath} (1)

which can be defined
\ln x \equiv \int_1^x {dt\over t}
\end{displaymath} (2)

for $x > 0$. The natural logarithm can also be defined for Complex Numbers as
\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
{d\over dx}\ln x={1\over x},
\end{displaymath} (4)

whereas logarithms in other bases have the more complicated Derivative
{d\over dx}\log_b x={1\over x\ln b}.
\end{displaymath} (5)

The Mercator Series

\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

\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)

\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 $z$, the natural logarithm satisfies

\ln z = \ln[re^{i(\theta +2n\pi)}] = \ln r+i(\theta +2n\pi)
\end{displaymath} (9)

PV(\ln z) = \ln r+i\theta,
\end{displaymath} (10)

where $PV$ is the Principal Value.

Some special values of the natural logarithm are

\ln 1 =0
\end{displaymath} (11)

\ln 0 =-\infty
\end{displaymath} (12)

\ln(-1)=\pi i
\end{displaymath} (13)

\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

(\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 $i>6$) (Bailey et al. 1995, Bailey and Plouffe).

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


Bailey, D.; Borwein, P.; and Plouffe, S. ``On the Rapid Computation of Various Polylogarithmic Constants.''

Bailey, D. and Plouffe, S. ``Recognizing Numerical Constants.''

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© 1996-9 Eric W. Weisstein