Question

Find the general value of ${log}_3$(3i). Where i= $\sqrt{-1}$

• A.
• B. = 2 + n I
• C. = 3 + n I
• D. = 1 + n I

Answer 1

The correct option is A

Given Let z = ${log}_3$(3i) = $\frac{{log}_e\left(3i\right)}{{log}_{e^3}}$

Convert 3i in the polar form

So write the given complex number in Euler's form.

= $\frac{1}{{log}_{e^3}}$ $\left[{log}_e(3.e^{i\frac{\pi }{2}})\right]$

= $\frac{1}{{log}_{e^3}}$ $[{log}_{e}3+{log}_e{e}^{i\frac{\pi }{2}}]$ = ${\frac{1}{log}}_{e^3}[{log}_{e}3+\frac{i\pi }{2}+2n\pi i]$ n ∈ I

= 1 + $\frac{1}{{log}_{e^3}}(\frac{i\pi }{2}+2n\pi i)$ n $ϵ$ I