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I have a problem that says:

Express $e^{\operatorname{Log}(3+4i)}$ in the form $x+iy$.

However, in class we usually only deal with natural log (this problem is from a textbook). How do I know when I can interpret ''Log'' as the natural log?

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    $\begingroup$ Once you're past calculus it's pretty much only used to mean natural log $\endgroup$ – Alex Mathers Feb 17 '16 at 18:57
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    $\begingroup$ Capital L $\operatorname{Log}$ usually has a very specific meaning when working with complex numbers, if I recall. $\endgroup$ – pjs36 Feb 17 '16 at 18:57
  • $\begingroup$ @pjs36 the capital just means to take the principle value of the argument. $\endgroup$ – whatwhatwhat Feb 17 '16 at 18:59
  • $\begingroup$ Yes but it's an extension of the natural log to the complex numbers (or to the Riemann surface associated with log z). $\endgroup$ – Bernard Feb 17 '16 at 19:05
  • $\begingroup$ Usually your textbook will standardize notation when it introduces logarithms, especially the capital logarithm that's used in complex analysis. Do not assume log is natural log when you look at other sources. In computer science it is often base 2, for example. $\endgroup$ – Alex R. Feb 17 '16 at 19:09
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Since $z\mapsto e^z$ is periodic with period $2\pi i$, it is not one-to-one, and so its inverse is a "multiple-valued function" (and so not really a "function" as textbooks define that term over the past hundred years or so). Thus $$ \operatorname{Log}(3+4i) = \log\sqrt{3^2+4^2} + i (\theta_0 + 2\pi n) $$ and $\theta_0 + 2\pi n$ is any of the angels in the first quadrant whose tangent is $4/3$.

Exponentiating this is exponentiating any of the complex number that when exponentiated yield $3+4i$. Hence $$ e^{\operatorname{Log}(3+4i)} = 3+4i. $$

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