# What calculate ln &#x2061;<!-- ⁡ --> i I would like to know how to calculate ln

What calculate $\mathrm{ln}i$
I would like to know how to calculate $\mathrm{ln}i$. I found a formula on the internet that says
$\mathrm{ln}z=\mathrm{ln}|z|+i\mathrm{Arg}\left(z\right)$
Then $|i|=1$ and $\mathrm{Arg}\left(i\right)$ is?
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Asdrubali2r
If you want to know what is $\mathrm{ln}i$ you should make yourself aware of what the result should be. By definition the logarithm of $i$ should be some complex number $z$ such that ${e}^{z}=i$. But by Euler's formula ${e}^{i\pi /2}=\mathrm{cos}\left(\pi /2\right)+i\mathrm{sin}\left(\pi /2\right)=i$, so you could say that "$\mathrm{ln}i=i\pi /2$". And that is true if we choose $\mathrm{ln}$ to be the principal branch of the complex logarithm.
But be aware that since the exponential function is periodic, also ${e}^{i\pi /2+2\pi ik}=i$ holds for all $k\in \mathbb{Z}$. Therefore the logarithm is a multi-valued function.
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Bruno Pittman
Given any non-zero complex number $z,$$\mathrm{Arg}\left(z\right)$ (the principal argument of $z$) is the unique $\theta \in \left(-\pi ,\pi \right]$ such that $z=|z|{e}^{i\theta }.$ Observing that $\theta$ gives a radian measure of rotation from the positive real axis, what must the principal argument of $i$ be, that is, how much must we rotate $|i|=1$ to get it to $i$ again?