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i notice that there are $e^{i\theta}$ in math,so is there a similar function in complex number system corresponding to logarithim in real number system?

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    Yes there is, but it is multivalued. See [here](http://en.wikipedia.org/wiki/Complex_logarithm)2012-04-22

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Yes. See the Wikipedia page on the complex logarithm.

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There is a problem because $(z\mapsto e^z) : \mathbb{C} \to \mathbb{C}$ is not an injection, e.g. $e^0 = 1 = e^{2\pi i}$. You can define a relation $R$ such that $z_1 R z_2 \iff z_2 = e^{z_1}$, but this relation would not be a function. Using the axiom of choice you could make it a function, i.e. for pick one element from each inverse image set and in fact mathematicians already did such thing: there is something known as principal value or principal branch. As others pointed out, there is nice wikipedia page, but you could also try mathworld and of course google.

Best luck!