Question

${\log }_{\frac{\pi }{4}}(-1+\sqrt{3}i)$ can be expressed in cartesian form as

• A. $\frac{\ln 2+\left(\frac{2\pi }{3}\right)i}{\ln \frac{\pi }{4}}$
• B. $\frac{\ln 2-\left(\frac{2\pi }{3}\right)i}{\ln \frac{\pi }{4}}$
• C. $\frac{\ln 2+\left(\frac{\pi }{3}\right)i}{\ln \frac{\pi }{2}}$
• D. $\frac{\ln 2-\left(\frac{2\pi }{3}\right)i}{\ln \frac{\pi }{2}}$

Answer 1

The correct option is A $\frac{\ln 2+\left(\frac{2\pi }{3}\right)i}{\ln \frac{\pi }{4}}$

We know that,
$\ln \left(z\right)=\ln \left(|z|\right)+i\arg \left(z\right)$

For $z=-1+\sqrt{3}i;|z|=2,\arg \left(z\right)=\pi -\frac{\pi }{3}=\frac{2\pi }{3}$

$\therefore {\log }_{\frac{\pi }{4}}(-1+\sqrt{3}i)=\frac{\ln (-1+\sqrt{3}i)}{\ln \frac{\pi }{4}}=\frac{\ln 2+\frac{2\pi }{3}i}{\ln \frac{\pi }{4}}$