Complex Logging

Algebra Level 1

ln ⁡ ( i ) = ? \large \ln {( i) } = \ ?

e i {e}^{i} 1 2 i π \frac{1}{2}i\pi Undefined π \pi

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Michael Fuller by Michael Fuller , 22, United Kingdom

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2 solutions

Michael Fuller
Sep 6, 2015

Let ln ⁡ i = x \ln {i} = x . Then i = e x i={e}^{x} , and i 2 = e 2 x = − 1 {i}^{2}={e}^{2x}=-1 .

We know that e i π = − 1 {e}^{i \pi}=-1 and e 2 x = − 1 ⇒ i π = 2 x {e}^{2x}=-1 \Rightarrow i \pi=2x .

Therefore ln ⁡ i = 1 2 i π \ln {i}=\large\color{#20A900}{\boxed{\frac{1}{2}i \pi}}

23 Helpful 2 Interesting 3 Brilliant 0 Confused

beautiful solution!!

Abu Bakar Khan - 5 years, 8 months ago

This is on the main branch of the log function... It is really undefined

Noam Pirani - 5 years, 2 months ago

The answer includes all the numbers of the form ( 4 n + 1 ) π 2 \dfrac{(4n+1)\pi}{2}

A Former Brilliant Member - 5 years, 3 months ago

5 Helpful 0 Interesting 0 Brilliant 0 Confused

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