Question

$\log (-1+\sqrt{3}i)$ can be expressed in cartesian form as

• A. $\ln 2-\left(\frac{2\pi }{3}\right)i$
• B. $\ln 2+\left(\frac{2\pi }{3}\right)i$
• C. $\ln 2-\left(\frac{\pi }{3}\right)i$
• D. $\ln 2+\left(\frac{\pi }{3}\right)i$

Answer 1

The correct option is B $\ln 2+\left(\frac{2\pi }{3}\right)i$

$\log \left(z\right)=\ln \left(|z|\right)+i\arg \left(z\right)$

For $z=-1+\sqrt{3}i,|z|=2$ and
$\tan \alpha =\left|\frac{\sqrt{3}}{-1}\right|$
$\Rightarrow \alpha =\frac{\pi }{3}$
$\therefore \arg \left(z\right)=\theta =\pi -\frac{\pi }{3}=\frac{2\pi }{3}$

Hence, $\log (-1+\sqrt{3}i)=\ln 2+\left(\frac{2\pi }{3}\right)i$