Digital Library of Mathematical Functions
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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.8 Identities

Contents

§4.8(i) Logarithms

In (4.8.1)–(4.8.4) z1z20.

4.8.1 Ln(z1z2)=Lnz1+Lnz2.

This is interpreted that every value of Ln(z1z2) is one of the values of Lnz1+Lnz2, and vice versa.

4.8.2 ln(z1z2)=lnz1+lnz2,
-πphz1+phz2π,
4.8.3 Lnz1z2=Lnz1-Lnz2,
4.8.4 lnz1z2=lnz1-lnz2,
-πphz1-phz2π.

In (4.8.5)–(4.8.7) and (4.8.10) z0.

4.8.5 Ln(zn)=nLnz,
n,
4.8.6 ln(zn)=nlnz,
n, -πnphzπ,
4.8.7 ln1z=-lnz,
|phz|π.
4.8.8 Ln(expz)=z+2kπ,
k,
4.8.9 ln(expz)=z,
-πzπ,
4.8.10 exp(lnz)=exp(Lnz)=z.

If a0 and az has its general value, then

4.8.11 Ln(az)=zLna+2kπ,
k.

If a0 and az has its principal value, then

4.8.12 ln(az)=zlna+2kπ,

where the integer k is chosen so that (-zlna)+2kπ[-π,π].

4.8.13 ln(ax)=xlna,
a>0.

§4.8(ii) Powers

4.8.14 az1az2 =az1+z2,
4.8.15 azbz =(ab)z,
-πpha+phbπ,
4.8.16 z1z2 =z1+z2,
4.8.17 (z1)z2 =z1z2,
-πz1π.

The restriction on z1 can be removed when z2 is an integer.