Question

For $z\ne 0$, define $\log z=\log |z|+i\left(argz\right)$

where $-\pi <arg\left(z\right)\le \pi$

i.e. $arg\left(z\right)$ stands for the principal argument of $z$. $\log z=1$ if and only if $z$ equals

• A. $e$
• B. $e,-e$
• C. $e+2k\pi i,k\in I$
• D. $\pm e+2k\pi i,k\in I$

Answer 1

The correct option is A $e$

$log\left(z\right)=1$
$log\left(e^{i\theta }\right)=1$
$i\theta =1$

Hence $z=e^{i\theta }$
$=e^1$
$=e$