Logarithm of a Complex Number

Q.

Find the value of i.$i^i$ is _______.

1. 

2. 

3. 

4. 

Q.

If z = $i^i$ . Express ${log}_e$z in A+iB form.Find the value of A and B.

1. A = , B =

2. A = 0 , B =

3. A = , B = 0

4. A = , B =

Q.

What is the value of sin(${log}_e{i}^i)$ ?

1. 0

2. 1

3. -1

4. 2

Q.

The locus of z satisfying the inequality $lo{g}_{\frac{1}{3}}$|z+1| > $lo{g}_{\frac{1}{3}}$|z-1| is

1. None of these

2. R(z)<0

3. R(z)>0

4. I(z)<0

Q.

Find the general value of ${log}_e$(i).

1. n I

2. n I

3. n I

4. n I

Q. The expression $\tan \left(i{\log }_e\left(\frac{2-3i}{2+3i}\right)\right)$ is equal to

1. $-\frac{12}{3}$
2. $-\frac{3}{2}$
3. $-\frac{12}{5}$
4. $0$

Q. The value of ${\log }_2(-3)$ is

1. $\frac{\log 3+2i\pi }{\log 2}$
2. $\frac{\log 2-i\pi }{\log 3}$
3. $\frac{\log 3+i\pi }{\log 2}$
   
4. $\frac{\log 2+i\pi }{\log 3}$

Q. The value of $\log \left(i^i\right)$ is

1. $-\frac{\pi }{2}$
2. $\frac{\pi }{2}$
3. $-\pi$
4. $\pi$

Q. The general value of $\log (1+i)+\log (1-i)$ is

1. $\log 2+i2k\pi ,k\in Z$
2. $\log \sqrt{2}+i2k\pi ,k\in Z$
3. $2\log 2+i2k\pi ,k\in Z$
4. $\log 3+i2k\pi ,k\in Z$

Q. If $z=(1-i{)}^{-i}$, then the correct option(s) is/are

1. $Re\left(z\right)=e^{-\frac{\pi }{4}}\cos \left(\log 2\right)$
2. $Im\left(z\right)=-e^{-\frac{\pi }{4}}\sin \left(\frac{1}{2}\log 2\right)$
3. $Im\left(z\right)=-e^{-\frac{\pi }{4}}\sin \left(\log 2\right)$
4. $Re\left(z\right)=e^{-\frac{\pi }{4}}\cos \left(\frac{1}{2}\log 2\right)$