Question

If z = (1 + i $\sqrt{3}$).Find the value of ${log}_e$z.

• A. In2 + i
• B. In2 + i
• C. In3 + i
• D. In3 + i

Answer 1

The correct option is A

In2 + i

z = (1 + i $\sqrt{3}$)

Polar form of z = r (cos$\theta$ + i sin$\theta$)

$\left|z\right|$ = $\sqrt{1+3}$ = 2

Arz(z) = $ta{n}^{-1}\left(\frac{\sqrt{3}}{1}\right)$ = $\frac{\pi }{3}$

z = 2 $(cos\frac{\pi }{3}+isin\frac{\pi }{3})$

Euler's form z = 2 $e^{i\frac{\pi }{3}}$

Take log on Euler's form of complex number.

${log}_e$z = ${log}_e$2 + ${log}_e{e}^{i\frac{\pi }{3}}$

= ${log}_e$2 + $\frac{i\pi }{3}$