- #1

Terrell

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## Homework Statement

Find ##z## in ##z^{1+i}=4##. Is my solution correct

## Homework Equations

##\log(z_1 z_2)=\log(z_1)+\log(z_2)## such that ##z_1, z_2\in \{z\in\Bbb{C} : (z=x+iy) \land (x\in\Bbb{R}) \land -\infty \lt y \lt +\infty\}##

##re^{i\theta}=r(\cos\theta + i\sin\theta)##

## The Attempt at a Solution

Note ##4=4(\cos 0 +i\sin 0)##. Thus,

\begin{align}

r^{1+i}e^{i\theta(1+i)}&=r^{1+i}e^{-\theta}(\cos\theta + i\sin\theta)=4(\cos 0 +i\sin 0)=4\

r^{1+i}e^{-\theta}&=4 \quad \text{and}\quad \theta=0\\

\log_e(r^{1+i}e^{-\theta})&=\log_{e}(4)\\

\log_(r^{1+i})+\log(e^{-\theta})&=\log_{e}(4)\\

(1+i)\log_{e}(r)-\theta&=\log_{e}(4)\\

\log_{e}(r)&=\frac{\log_e(4)}{1+i}\\

\log_{e}(r)&=\log_{e}(4^{\frac{1}{1+i}})\\

r&=4^{\frac{1}{1+i}}

\end{align}