# §4.12 Generalized Logarithms and Exponentials

A generalized exponential function $\phi(x)$ satisfies the equations

 4.12.1 $\displaystyle\phi(x+1)$ $\displaystyle=e^{\phi(x)},$ $-1, ⓘ Symbols: $\mathrm{e}$: base of exponential function, $x$: real variable and $\phi(z)$: generalized exponential Permalink: http://dlmf.nist.gov/4.12.E1 Encodings: TeX, pMML, png See also: Annotations for 4.12 and 4 4.12.2 $\displaystyle\phi(0)$ $\displaystyle=0,$ ⓘ Symbols: $\phi(z)$: generalized exponential Permalink: http://dlmf.nist.gov/4.12.E2 Encodings: TeX, pMML, png See also: Annotations for 4.12 and 4

and is strictly increasing when $0\leq x\leq 1$. Its inverse $\psi(x)$ is called a generalized logarithm. It, too, is strictly increasing when $0\leq x\leq 1$, and

 4.12.3 $\displaystyle\psi(e^{x})$ $\displaystyle=1+\psi(x),$ $-\infty, ⓘ Symbols: $\mathrm{e}$: base of exponential function, $x$: real variable and $\psi(x)$: generalized logarithm Permalink: http://dlmf.nist.gov/4.12.E3 Encodings: TeX, pMML, png See also: Annotations for 4.12 and 4 4.12.4 $\displaystyle\psi(0)$ $\displaystyle=0.$ ⓘ Symbols: $\psi(x)$: generalized logarithm Permalink: http://dlmf.nist.gov/4.12.E4 Encodings: TeX, pMML, png See also: Annotations for 4.12 and 4

These functions are not unique. The simplest choice is given by

 4.12.5 $\phi(x)=\psi(x)=x,$ $0\leq x\leq 1$. ⓘ Symbols: $x$: real variable, $\phi(z)$: generalized exponential and $\psi(x)$: generalized logarithm Permalink: http://dlmf.nist.gov/4.12.E5 Encodings: TeX, pMML, png See also: Annotations for 4.12 and 4

Then

 4.12.6 $\phi(x)=\ln\left(x+1\right),$ $-1, ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $x$: real variable and $\phi(z)$: generalized exponential Permalink: http://dlmf.nist.gov/4.12.E6 Encodings: TeX, pMML, png See also: Annotations for 4.12 and 4

and

 4.12.7 $\phi(x)=\exp\exp\cdots\exp\left(x-\left\lfloor x\right\rfloor\right),$ $x>1$,

where the exponentiations are carried out $\left\lfloor x\right\rfloor$ times. Correspondingly,

 4.12.8 $\psi(x)=e^{x}-1,$ $-\infty, ⓘ Symbols: $\mathrm{e}$: base of exponential function, $x$: real variable and $\psi(x)$: generalized logarithm Permalink: http://dlmf.nist.gov/4.12.E8 Encodings: TeX, pMML, png See also: Annotations for 4.12 and 4

and

 4.12.9 $\psi(x)=\ell+{\ln^{(\ell)}}x,$ $x>1$, ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $x$: real variable, $\psi(x)$: generalized logarithm and $\ell$: positive integer Permalink: http://dlmf.nist.gov/4.12.E9 Encodings: TeX, pMML, png See also: Annotations for 4.12 and 4

where ${\ln^{(\ell)}}x$ denotes the $\ell$-th repeated logarithm of $x$, and $\ell$ is the positive integer determined by the condition

 4.12.10 $0\leq{\ln^{(\ell)}}x<1.$ ⓘ Defines: $\ell$: positive integer (locally) Symbols: $\ln\NVar{z}$: principal branch of logarithm function and $x$: real variable Permalink: http://dlmf.nist.gov/4.12.E10 Encodings: TeX, pMML, png See also: Annotations for 4.12 and 4

Both $\phi(x)$ and $\psi(x)$ are continuously differentiable.

For further information, see Clenshaw et al. (1986). For $C^{\infty}$ generalized logarithms, see Walker (1991). For analytic generalized logarithms, see Kneser (1950).