# Complex Numbers Questions and Answers – Logarithm of Complex Numbers

This set of Complex Analysis Questions and Answers for Campus interviews focuses on “Logarithm of Complex Numbers”.

1. Find the value of log⁡(-6).
a) log6+2iπ
b) log⁡36+iπ
c) log6+2iπ
d) log6+iπ

Explanation: We know that
$$log⁡(x-iy)=\frac{1}{2} log⁡(x^2+y^2)+itan^{-1} (\frac{y}{x})$$
Putting x=-6 and y=0.
$$log⁡(-6)=\frac{1}{2} log⁡(36)+itan^{-1} (\frac{0}{-6})$$
$$log⁡(-6)=log6+iπ$$.

2. Find the value of log2(-3).
a) $$\frac{log⁡_3+i8\pi}{log_2}$$
b) $$\frac{log⁡_3+3i\pi}{log_2}$$
c) $$\frac{log⁡_3+i\pi}{log_2}$$
d) $$\frac{log_⁡2+i\pi}{log_3}$$

Explanation: In this problem, we change the base to e
$$log_2(-3)=\frac{log_e(-3)}{loge(2)}$$
$$log_2(-3)=\frac{log⁡_3+i\pi}{log_2}$$.

3. Represent ii in terms of e.
a) $$e^{\frac{-\pi}{3}}$$
b) $$e^{\frac{-3\pi}{2}}$$
c) $$e^{\frac{-\pi}{2}}$$
d) $$e^{\frac{-\pi}{6}}$$

Explanation: We know that
$$a^x=e^{x loga}$$
$$i^i=e^{i log⁡i}$$
We also know from the definition of logarithm,
$$log⁡i=\frac{i\pi}{2}$$
$$i^i=e^{i(\frac{i\pi}{2})}=e^{\frac{-\pi}{2}}$$.

Sanfoundry Global Education & Learning Series – Complex Analysis.

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