# §4.8 Identities

## §4.8(i) Logarithms

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

 4.8.1 $\mathop{\mathrm{Ln}\/}\nolimits\!\left(z_{1}z_{2}\right)=\mathop{\mathrm{Ln}\/% }\nolimits z_{1}+\mathop{\mathrm{Ln}\/}\nolimits z_{2}.$ Symbols: $\mathop{\mathrm{Ln}\/}\nolimits\NVar{z}$: 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)

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

 4.8.2 $\mathop{\ln\/}\nolimits\!\left(z_{1}z_{2}\right)=\mathop{\ln\/}\nolimits z_{1}% +\mathop{\ln\/}\nolimits z_{2},$ $-\pi\leq\mathop{\mathrm{ph}\/}\nolimits z_{1}+\mathop{\mathrm{ph}\/}\nolimits z% _{2}\leq\pi$,
 4.8.3 $\mathop{\mathrm{Ln}\/}\nolimits\frac{z_{1}}{z_{2}}=\mathop{\mathrm{Ln}\/}% \nolimits z_{1}-\mathop{\mathrm{Ln}\/}\nolimits z_{2},$ Symbols: $\mathop{\mathrm{Ln}\/}\nolimits\NVar{z}$: 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)
 4.8.4 $\mathop{\ln\/}\nolimits\frac{z_{1}}{z_{2}}=\mathop{\ln\/}\nolimits z_{1}-% \mathop{\ln\/}\nolimits z_{2},$ $-\pi\leq\mathop{\mathrm{ph}\/}\nolimits z_{1}-\mathop{\mathrm{ph}\/}\nolimits z% _{2}\leq\pi$. Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$: 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)

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

 4.8.5 $\mathop{\mathrm{Ln}\/}\nolimits\!\left(z^{n}\right)=n\mathop{\mathrm{Ln}\/}% \nolimits z,$ $n\in\mathbb{Z}$, Symbols: $\in$: element of, $\mathbb{Z}$: set of all integers, $\mathop{\mathrm{Ln}\/}\nolimits\NVar{z}$: 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)
 4.8.6 $\mathop{\ln\/}\nolimits\!\left(z^{n}\right)=n\mathop{\ln\/}\nolimits z,$ $n\in\mathbb{Z}$, $-\pi\leq n\mathop{\mathrm{ph}\/}\nolimits z\leq\pi$,
 4.8.7 $\mathop{\ln\/}\nolimits\frac{1}{z}=-\mathop{\ln\/}\nolimits z,$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$.
 4.8.8 $\mathop{\mathrm{Ln}\/}\nolimits\!\left(\mathop{\exp\/}\nolimits z\right)=z+2k% \pi\mathrm{i},$ $k\in\mathbb{Z}$,
 4.8.9 $\mathop{\ln\/}\nolimits\!\left(\mathop{\exp\/}\nolimits z\right)=z,$ $-\pi\leq\Im{z}\leq\pi$,
 4.8.10 $\mathop{\exp\/}\nolimits\!\left(\mathop{\ln\/}\nolimits z\right)=\mathop{\exp% \/}\nolimits\!\left(\mathop{\mathrm{Ln}\/}\nolimits z\right)=z.$ Symbols: $\mathop{\exp\/}\nolimits\NVar{z}$: exponential function, $\mathop{\mathrm{Ln}\/}\nolimits\NVar{z}$: general logarithm function, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 4.2.4 Referenced by: §4.8(i) Permalink: http://dlmf.nist.gov/4.8.E10 Encodings: TeX, pMML, png See also: Annotations for 4.8(i)

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

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

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

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

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

 4.8.13 $\mathop{\ln\/}\nolimits\!\left(a^{x}\right)=x\mathop{\ln\/}\nolimits a,$ $a>0$. Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: 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(ii) Powers

 4.8.14 $\displaystyle a^{z_{1}}a^{z_{2}}$ $\displaystyle=a^{z_{1}+z_{2}},$ Symbols: $a$: real or complex constant and $z$: complex variable A&S Ref: 4.2.15 Permalink: http://dlmf.nist.gov/4.8.E14 Encodings: TeX, pMML, png See also: Annotations for 4.8(ii) 4.8.15 $\displaystyle a^{z}b^{z}$ $\displaystyle=(ab)^{z},$ $-\pi\leq\mathop{\mathrm{ph}\/}\nolimits a+\mathop{\mathrm{ph}\/}\nolimits b\leq\pi$, Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\mathrm{ph}\/}\nolimits$: 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) 4.8.16 $\displaystyle e^{z_{1}}e^{z_{2}}$ $\displaystyle=e^{z_{1}+z_{2}},$ Symbols: $\mathrm{e}$: base of exponential function 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) 4.8.17 $\displaystyle(e^{z_{1}})^{z_{2}}$ $\displaystyle=e^{z_{1}z_{2}},$ $-\pi\leq\Im{z_{1}}\leq\pi$. Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $\Im{}$: 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)

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