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

§4.8 Identities

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

§4.8(i) Logarithms

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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

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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

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§4.8 Identities

Contents

§4.8(i) Logarithms

§4.8(ii) Powers