4 Elementary Functions Logarithm, Exponential, Powers 4.7 Derivatives and Differential Equations 4.9 Continued Fractions

§4.8 Identities

ⓘ

Permalink:: http://dlmf.nist.gov/4.8
See also:: Annotations for Ch.4

§4.8(i) Logarithms

ⓘ

Keywords:: identities, logarithm function
Notes:: See Levinson and Redheffer (1970, pp. 62–66).
Permalink:: http://dlmf.nist.gov/4.8.i
See also:: Annotations for §4.8 and Ch.4

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		$\ln\left(z^{n}\right)=n\ln z,$
$n\in\mathbb{Z}$ , $-\pi\leq n\operatorname{ph}z\leq\pi$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathbb{Z}$ : set of all integers, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $n$ : integer and $z$ : complex variable A&S Ref: 4.1.11 Permalink: http://dlmf.nist.gov/4.8.E6 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4

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		$\operatorname{Ln}\left(\exp z\right)=z+2k\pi\mathrm{i},$
$k\in\mathbb{Z}$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $k$ : integer and $z$ : complex variable A&S Ref: 4.2.2 Permalink: http://dlmf.nist.gov/4.8.E8 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4

4.8.9		$\ln\left(\exp z\right)=z,$
$-\pi\leq\Im z\leq\pi$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\Im$ : imaginary part, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 4.2.3 Permalink: http://dlmf.nist.gov/4.8.E9 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4

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		$\operatorname{Ln}\left(a^{z}\right)=z\operatorname{Ln}a+2k\pi\mathrm{i},$
$k\in\mathbb{Z}$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $k$ : integer, $a$ : real or complex constant and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.8.E11 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 principal value, then

4.8.12		$\ln\left(a^{z}\right)=z\ln a+2k\pi\mathrm{i},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : integer, $a$ : real or complex constant and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.8.E12 Encodings: TeX, pMML, png See also: Annotations for §4.8(i), §4.8 and Ch.4

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

ⓘ

Keywords:: exponential function, identities, power function
Notes:: See Hobson (1928, pp. 297–299).
Permalink:: http://dlmf.nist.gov/4.8.ii
See also:: Annotations for §4.8 and Ch.4

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.

4.7 Derivatives and Differential Equations 4.9 Continued Fractions

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§4.8 Identities

Contents

§4.8(i) Logarithms

§4.8(ii) Powers

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!