Natural log of a negative number ln⁡(−1)=ln⁡(eiπ)=iπ

derlingasmh

derlingasmh

Answered

2022-01-25

Natural log of a negative number
ln(1)=ln(eiπ)=iπ

Answer & Explanation

Tapanuiwp

Tapanuiwp

Expert

2022-01-26Added 13 answers

Context is important here. In the context of real numbers, negative numbers have no logarithms (and neither does 0) because log(x) is a number y such that ey=x and ey is always greater than 0.
On the other hand, in the context of complex numbers, every complex number other than 0 has logarithms. In fact, any such complex number has infinitely many logarithms! You are right when you claim that iπ is a logarithm of 1. However, every complex number of the form πi+2π (with nZ) is also a logarithm of 1, since
eπi+2nπi=eπie2π=(1)×1=1.

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