# How to separate out real and imaginary terms

I have an equation like this: $$a+ib = \log(x+iy).$$

I need to separate the real and imaginary part in RHS so that I can equate the real part of LHS to real part of RHS and imaginary to imaginary part of RHS.

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You could convert the complex number from algebraic to polar form $r\cdot e^{j\theta}$ where $r=\sqrt{x^2+y^2}$ and $\theta=\arctan\left(\frac{y}{x}\right)$

After that, you can split the $\log$ function into real and imaginary parts:

$$\log(r\cdot e^{j\theta}) =\log(r) +j\theta\log(e)$$ $$= \log(r) +j\theta$$ (assuming natural logarithm)

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$$e^{a+jb}=a+jb\\ e^ae^{jb}=a+jb\\ e^a(\cos(b)+j\sin(b))=a+jb$$ Then real and imaginary parts have been separated.
That I know but how do I seperate it in real and imaginary part when the complex number is inside a log funciton?. In the above equation forget the LHS. I need to solve the RHS $\log$$x+jy$$$ and seperate this into the form a + jb –  Shivji Jun 14 at 3:53