Logarithm

Logarithm is holomorphic function, inverse of the exponential.

For the natural exponential, id est, exponential to base e, the logarithm is called "natural" and is denoted with symbol \(\ln\) in mathematical notations; id est,
 * \( \exp(\ln(z))=z ~ ~\) for all from the range of definition;

Logarithm is defined for any complex argument, except zero and negative values.

In languages C++, Fortran, Matlab, Maple, the natural logarithm of variable \(x\) is denoted with
 * \(\log(x)\)

In Mathematica, it is denoted with
 * \(\mathrm{Log}[x]\)

Various values of base
By default, the base of logarithm is assumed to be \(\mathrm{e}\!=\exp(1)\approx 2.71\) ..

For arbitrary base \(b\), the function \(\log_b\) is inverse function of the exponential to base \(b\);, id est, \(\exp_b\);
 * \(\exp_b(z)=b^z\)

Usually, it is assumed that the base \(b\) is positive number larger than unity. If the base is not \(\mathrm e\), then in the mathematical notations the symbol "log" is used; the base is indicated as subscript:
 * \(\log_b(x)\)

Logarithm to base \(b\) is Abel function of operation \(z\mapsto bz\); For such Abel function, the multiplication to constant \(b\) is considered as the Transfer funciton.

For complex values of base \(b\), it is convenient to express the logarithm to base \(b\) through the natural logarithm:

\log_b(z)=\ln(z)/\ln(b) \). This expression determines the cut lines of logarithm, whule the cut line of the natural logarithm is determined (along the real axis from zero to \(-\)infinity); the cut line of logarithm with respect to base is the same as that with respect to its argument.

Logarithm as transfer function
Let transfer function \(T=\log_b\). The superfunction \(F\) can be expresed through the Tetration tet:

\( F(z)=\mathrm{tet}_b(-z)\)

The corresponding Abel function \(G=F^{-1}\) can be expressed through the ArcTetration ate:

\( G(z)=-\mathrm{ate}_b(z)\)

The name of superfunction \(F\) above could be SuperLogarithm, but, unfortunately, this name is already used meaning ArcTetration. In order to avoid confusion, term SuperLogarithm should not be used at all, in any of its meanings.

Logarithm as superfunction
It is easy to define the transfer function \(T\) such that \(\log_b\) is its superfunction, let

\( T(z)=\log_b\Big( 1+b^z\Big)\)

Then, \(\log_b\) appears as its superfunction satisfying the transfer equation

\(T\Big(\log_b(z)\Big)=\log_b(z\!+\!1) \)

In this case, \(b^z\) appears as the Abel function.

Logarithm as Abel function
For the transfer function \(T\) such that \(T(z)=b z\), the exponential \(\exp_b\) appear as superfunction, satisfying the transfer equation

\( T\Big( \exp_b(z)) = \exp_b(1\!+\!z)\)

and the inverse function \(\log_b=\exp_b^{-1}\) appears as the Abel function, satisfying the Abel equation

\( \log_b\Big( T(z) \Big) = 1+\log_b(z)\)

Derivation and integration
\(\displaystyle {\log_b}^{\prime}(z)=\frac{1}{z\, \ln(b)}\)

In particular, for \(b\!=\!\mathrm e\), we have \(\log_b=\ln\), and \(\ln(b)\!=\!1\), so

\(\displaystyle {\ln}^{\prime}(z)=\frac{1}{z}\)

The integration can be expressed with

\(\displaystyle \int_1^z \log_b(t) ~\mathrm d t= \Big(1-z + z \ln(z)\Big)/\ln(b)= \frac{1\!-\!z}{\ln(b)} + z\, \log_b(z)\)

And, in particular, for \(b\!=\!\mathrm e\), we have \(\log_b=\ln\), and \(\ln(b)\!=\!1\), so

\(\displaystyle \int_1^z \ln(t) ~\mathrm d t= 1-z + z\, \ln(z)\)

Fixed points of logarithm
Fixed points of \(\log_b\) can be expressed with function Filog:
 * \(L_1=\mathrm{Filog}(a)\)
 * \(L_2=\mathrm{Filog}(a^*)^*\)

where \(a=\ln(b)\).

For positive values of the imaginary part of argument, the fixed point of logarithm can be expressed also through the LambertW function:
 * \( L=-\mathrm{LamberW}(-a)/a\)

where, again, \(a=\ln(b)\).

Keywords
Elementary function, Filog, Exponent, tetration