We have an answer to "Natural log of a negative number ln⁡(−1)=ln⁡(eiπ)=iπ"

High School
Algebra
Pre-Algebra
Negative numbers and coordinate plane

derlingasmh

Answered question

2022-01-25

Natural log of a negative number

\ln (- 1) = \ln (e^{i π}) = i π

Answer & Explanation

Tapanuiwp

Beginner2022-01-26Added 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 $e^{y} = x$ and $e^{y}$ 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 $i π$ is a logarithm of $- 1$ . However, every complex number of the form $π i + 2 π \in$ (with $n \in Z$ ) is also a logarithm of $- 1$ , since
$e^{π i + 2 n π i} = e^{π i} e^{2 π \in} = (- 1) \times 1 = - 1 .$

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