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Showing content from https://dlmf.nist.gov/4.8 below:

§4.8 Identities ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions

§4.8 Identities Contents
  1. §4.8(i) Logarithms
  2. §4.8(ii) Powers
§4.8(i) Logarithms

In (4.8.1)–(4.8.4) z 1 ⁢ z 2 ≠ 0 .

This is interpreted that every value of Ln ⁡ ( z 1 ⁢ z 2 ) is one of the values of Ln ⁡ z 1 + Ln ⁡ z 2 , and vice versa.

In (4.8.5)–(4.8.7) and (4.8.10) z ≠ 0 .

4.8.6 ln ⁡ ( zn ) = n ⁢ ln ⁡ z , n ∈ ℤ , − π ≤ n ⁢ ph ⁡ z ≤ π , 4.8.8 Ln ⁡ ( exp ⁡ z ) = z + 2 ⁢ k ⁢ π ⁢ i , k ∈ ℤ ,

If a ≠ 0 and az has its general value, then

4.8.11 Ln ⁡ ( az ) = z ⁢ Ln ⁡ a + 2 ⁢ k ⁢ π ⁢ i , k ∈ ℤ .

If a ≠ 0 and az has its principal value, then

where the integer k is chosen so that ℜ ⁡ ( − i ⁢ z ⁢ ln ⁡ a ) + 2 ⁢ k ⁢ π ∈ [ − π , π ] .

§4.8(ii) Powers 4.8.14 a z 1 ⁢ a z 2 = a z 1 + z 2 , a ≠ 0 , 4.8.15 az ⁢ bz = ( a ⁢ b ) z , − π ≤ ph ⁡ a + ph ⁡ b ≤ π , 4.8.16 e z 1 ⁢ e z 2 = e z 1 + z 2 , 4.8.17 ( e z 1 ) z 2 = e z 1 ⁢ z 2 , − π ≤ ℑ ⁡ z 1 ≤ π .

The restriction on z 1 can be removed when z 2 is an integer.

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