$$3^i=e^{i\ln(3)}.$$
Why is this equality true, could anyone demonstrate it?
There may be a place where it’s already been demonstrated.
I see a lot of the demos of $i^i$ but with real base transforming to $\ln(3)$ I didn’t find.
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1Check out Euler's Formula - https://en.wikipedia.org/wiki/Euler%27s_formula – Jamie Alizadeh Sep 24 '22 at 02:26
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6This is by definition. For $a$ a positive real number and $z$ complex we just define $a^z = e^{z \ln a}$. The nontrivial fact is that this is consistent with the usual definition of the exponential if $z$ is real, which is just a matter of applying logarithm and exponent rules. – Qiaochu Yuan Sep 24 '22 at 02:35
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4Does this answer your question? How to calculate a real number raised to an imaginary number You already have asked this before. – Dietrich Burde Sep 24 '22 at 09:19
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It has been mentioned , but agaian : It is very important that we usually run into trouble, if we use the exponential rules and compelx numbers are involved. Concerning , $i^i$ , it is not uniquely defined. We must choose a branch to establish a value, usually the branch is chosen such that we get a real number. – Peter Sep 24 '22 at 11:17