1. ## real and imaginary

Hi, can someone please help me write $\displaystyle Log [(1+i)^4]$ in terms of its real and imaginary parts?

2. write

$\displaystyle 1+i = \sqrt{2}\cdot (e^{i\frac{\pi}{4}})$

Look at it for some time
...do you need more help than that .?

3. omigosh i am so lost in this subject

coz i thought that i had to use $\displaystyle Log z = ln|z| + i Arg(z)$

well this is what i did, but like yeahh i really dont know what im doing
$\displaystyle ln|1+i|^4 + i Arg(1+i)^4$
$\displaystyle 4 ln|1+i| + i Arg(1+i)^4$

4. Convert to polars.

$\displaystyle x + iy = r(\cos{\theta} + i\sin{\theta}) = r\,e^{i\theta}$

You have $\displaystyle 1 + i$, which is in the first quadrant.

$\displaystyle r = \sqrt{1^2 + 1^2} = \sqrt{2}$.

$\displaystyle \theta = \arctan{\frac{1}{1}} = \frac{\pi}{4}$.

Therefore $\displaystyle 1 + i = \sqrt{2}\left(\cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}}\right) = \sqrt{2}\,e^{\frac{\pi}{4}i}$.

So $\displaystyle (1 + i)^4 = (\sqrt{2}\,e^{\frac{\pi}{4}i})^4$

$\displaystyle = 4\,e^{\pi i}$

$\displaystyle = -4$.

Can you see that

$\displaystyle \log{[(1 + i)^4]} = \log{(-4)}$

$\displaystyle = \ln{|-4|} + i\arg{(-4)}$

$\displaystyle = \ln{4} + \pi i$

$\displaystyle = 2\ln{2} + \pi i$.

Now it is written in terms of its real and imaginary parts.