Question

The value of $\log \left(i^i\right)$ is

• A. $-\frac{\pi }{2}$
• B. $\frac{\pi }{2}$
• C. $-\pi$
• D. $\pi$

Answer 1

The correct option is A $-\frac{\pi }{2}$

$\log \left(i^i\right)=i\log i[\because {\log }^{}{x}^n=n{\log }^{}x]$
$=i\cdot \log e^{\left(i\pi /2\right)}[\because i=\cos \left(\frac{\pi }{2}\right)+i\sin \left(\frac{\pi }{2}\right)=e^{i\pi /2}]$
$=i\cdot i\frac{\pi }{2}$
$=-\frac{\pi }{2}[\because i^2=-1]$