# §4.8 Identities

## §4.8(i) Logarithms

In (4.8.1)–(4.8.4) $z_{1}z_{2}\neq 0$.

 4.8.1 $\operatorname{Ln}\left(z_{1}z_{2}\right)=\operatorname{Ln}z_{1}+\operatorname{% Ln}z_{2}.$ ⓘ Symbols: $\operatorname{Ln}\NVar{z}$: general logarithm function and $z$: complex variable A&S Ref: 4.1.6 Referenced by: §4.8(i) Permalink: http://dlmf.nist.gov/4.8.E1 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4

This is interpreted that every value of $\operatorname{Ln}\left(z_{1}z_{2}\right)$ is one of the values of $\operatorname{Ln}z_{1}+\operatorname{Ln}z_{2}$, and vice versa.

 4.8.2 $\ln\left(z_{1}z_{2}\right)=\ln z_{1}+\ln z_{2},$ $-\pi\leq\operatorname{ph}z_{1}+\operatorname{ph}z_{2}\leq\pi$, ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: principal branch of logarithm function, $\operatorname{ph}$: phase and $z$: complex variable A&S Ref: 4.1.7 Permalink: http://dlmf.nist.gov/4.8.E2 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4
 4.8.3 $\operatorname{Ln}\frac{z_{1}}{z_{2}}=\operatorname{Ln}z_{1}-\operatorname{Ln}z% _{2},$ ⓘ Symbols: $\operatorname{Ln}\NVar{z}$: general logarithm function and $z$: complex variable A&S Ref: 4.1.8 Permalink: http://dlmf.nist.gov/4.8.E3 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4
 4.8.4 $\ln\frac{z_{1}}{z_{2}}=\ln z_{1}-\ln z_{2},$ $-\pi\leq\operatorname{ph}z_{1}-\operatorname{ph}z_{2}\leq\pi$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: principal branch of logarithm function, $\operatorname{ph}$: phase and $z$: complex variable A&S Ref: 4.1.9 Referenced by: §4.8(i) Permalink: http://dlmf.nist.gov/4.8.E4 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4

In (4.8.5)–(4.8.7) and (4.8.10) $z\neq 0$.

 4.8.5 $\operatorname{Ln}\left(z^{n}\right)=n\operatorname{Ln}z,$ $n\in\mathbb{Z}$, ⓘ Symbols: $\in$: element of, $\mathbb{Z}$: set of all integers, $\operatorname{Ln}\NVar{z}$: general logarithm function, $n$: integer and $z$: complex variable A&S Ref: 4.1.10 Referenced by: §4.8(i) Permalink: http://dlmf.nist.gov/4.8.E5 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4
 4.8.6 $\ln\left(z^{n}\right)=n\ln z,$ $n\in\mathbb{Z}$, $-\pi\leq n\operatorname{ph}z\leq\pi$,
 4.8.7 $\ln\frac{1}{z}=-\ln z,$ $|\operatorname{ph}z|\leq\pi$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: principal branch of logarithm function, $\operatorname{ph}$: phase and $z$: complex variable Referenced by: §4.2(i), §4.8(i) Permalink: http://dlmf.nist.gov/4.8.E7 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4
 4.8.8 $\operatorname{Ln}\left(\exp z\right)=z+2k\pi\mathrm{i},$ $k\in\mathbb{Z}$,
 4.8.9 $\ln\left(\exp z\right)=z,$ $-\pi\leq\Im z\leq\pi$,
 4.8.10 $\exp\left(\ln z\right)=\exp\left(\operatorname{Ln}z\right)=z.$ ⓘ Symbols: $\exp\NVar{z}$: exponential function, $\operatorname{Ln}\NVar{z}$: general logarithm function, $\ln\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 4.2.4 Referenced by: (25.10.2), §4.8(i) Permalink: http://dlmf.nist.gov/4.8.E10 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4

If $a\neq 0$ and $a^{z}$ has its general value, then

 4.8.11 $\operatorname{Ln}\left(a^{z}\right)=z\operatorname{Ln}a+2k\pi\mathrm{i},$ $k\in\mathbb{Z}$.

If $a\neq 0$ and $a^{z}$ has its principal value, then

 4.8.12 $\ln\left(a^{z}\right)=z\ln a+2k\pi\mathrm{i},$

where the integer $k$ is chosen so that $\Re\left(-\mathrm{i}z\ln a\right)+2k\pi\in[-\pi,\pi]$.

 4.8.13 $\ln\left(a^{x}\right)=x\ln a,$ $a>0$. ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $a$: real or complex constant and $x$: real variable Permalink: http://dlmf.nist.gov/4.8.E13 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4

## §4.8(ii) Powers

 4.8.14 $\displaystyle a^{z_{1}}a^{z_{2}}$ $\displaystyle=a^{z_{1}+z_{2}},$ $a\neq 0$, ⓘ Symbols: $a$: real or complex constant and $z$: complex variable A&S Ref: 4.2.15 Referenced by: Erratum (V1.0.20) for Equation (4.8.14) Permalink: http://dlmf.nist.gov/4.8.E14 Encodings: TeX, pMML, png Clarification (effective with 1.0.20): The constraint $a\neq 0$ was added. Suggested 2018-08-13 by Ted Ersek See also: Annotations for §4.8(ii), §4.8 and Ch.4 4.8.15 $\displaystyle a^{z}b^{z}$ $\displaystyle=(ab)^{z},$ $-\pi\leq\operatorname{ph}a+\operatorname{ph}b\leq\pi$, ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$: phase, $a$: real or complex constant and $z$: complex variable A&S Ref: 4.2.16 Permalink: http://dlmf.nist.gov/4.8.E15 Encodings: TeX, pMML, png See also: Annotations for §4.8(ii), §4.8 and Ch.4 4.8.16 $\displaystyle e^{z_{1}}e^{z_{2}}$ $\displaystyle=e^{z_{1}+z_{2}},$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm and $z$: complex variable A&S Ref: 4.2.18 Permalink: http://dlmf.nist.gov/4.8.E16 Encodings: TeX, pMML, png See also: Annotations for §4.8(ii), §4.8 and Ch.4 4.8.17 $\displaystyle(e^{z_{1}})^{z_{2}}$ $\displaystyle=e^{z_{1}z_{2}},$ $-\pi\leq\Im z_{1}\leq\pi$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\Im$: imaginary part and $z$: complex variable A&S Ref: 4.2.19 Permalink: http://dlmf.nist.gov/4.8.E17 Encodings: TeX, pMML, png See also: Annotations for §4.8(ii), §4.8 and Ch.4

The restriction on $z_{1}$ can be removed when $z_{2}$ is an integer.