Whilst going over old material, I remember a friend had told me something that was I found quite interesting a while ago about logarithms, and it's troubled me enough for the past few minutes for me to come to fellow mathematics enthusiasts for some insight regarding this conundrum.

If you were to enable imaginary numbers on a calculator, or even do this out by hand, it's found that:

\(ln { (-1) } =\pi i\)

I have a good understanding of logarithms to know what this technically *means*, but I'm curious as to why it is.

How come \({ e }^{ \pi i } = { (-1) }\) ? How does the ratio of the circumference of a circle to it's diameter and a complex number come into play in exponential and logarithmic functions?

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## Comments

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TopNewestKhanacademy teaches this part well. On how the Euler identity comes about. Look up a few of the videos.

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What all topics did you learn from Khan Academy ?

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idk... This was a long time ago, and one of the first thing i learnt from there. Afterwards I did a lot of organic chemistry. But that's it. Haven't logged onto it for many months.

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@Deepanshu Gupta ,@Raghav Vaidyanathan ,@Shashwat Shukla ,@Mvs Saketh

What are your views on a possible geometric interpretation ?

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Complex logarithms are actually periodic with period \(2\pi\)

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Hmm , complex numbers is not my forte too .

I'm sure you must be knowing that \(e^{ix} = \cos x + i \sin x\) . So replace \(x\) by \(\pi\) ,to get your answer .

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This is actually my first time hearing of that identity! Is that where the Eulurs number came from?

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Do you know MacLaurin's expansion? You may use it to get that identity.

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