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Generally for logarithms If I have $2^4=16$ then it means $\log_2(16)=4$ (Here 2 is the base)

So the value of logarithm basically tells us how many times to multiply the base for the number.

When we take ln it simply means base is e

Now begins my question

What is the logarithm of complex number?

I thought since logarithm tells us how many times to multiply the base. If I would take logarithm I would get real numbers Because no matter the number if real numbers are multiplied the answer is real.

But the book I have says its complex.It even has an derivation for it.

Can someone explain logarithms of complex number relating real or at least share some resources?

Thank you

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    $\begingroup$ Are you familiar with complex numbers in polar form? Ex $z = re^{i \theta}$? $\endgroup$ Commented Sep 21, 2020 at 12:55
  • $\begingroup$ Yes its $r*(cos(theta)+i*sin(theta)) $ $\endgroup$ Commented Sep 21, 2020 at 12:57
  • $\begingroup$ Then $\log z=\log r+i\theta$. $\endgroup$ Commented Sep 21, 2020 at 13:03
  • $\begingroup$ The complex logarithms are .. complex. Any number $w$ with $e^w = z$ is called a logarithm of $z$ and a number can have (infinitely) many logarithms. For a starter: en.wikipedia.org/wiki/Complex_logarithm. $\endgroup$ Commented Sep 21, 2020 at 13:05

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Notice that a complex number $z$ can be written as $$z=|z|e^{i\theta}$$

Where $|x|$ means the module of $x$

So the logarithim of $z$ would be

$$z=|z|e^{i\theta}\\ \log z=\log (|z|e^{i\theta})\\ \log z= \log{|z|+\log{e^{i\theta}}}\\ \log{z}=\log{|z|}+i\theta$$

For the calculation of $|z|$ we can simply use pitagorean theorem. For the calculation of $\theta$ we can use the $\arctan \theta$ since we know both the adjacent and opposite sides. Notice that $\theta$ is not only one solution, because $\tan{(\theta+2\pi n)}= \tan{\theta}$, so we actually get a branch of solutions. In despite of this fact, we usually just use the main one, which is the one belonging to the set $(-\pi,\pi]$

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