Question

Find the value of $\log (\log i)=$ and choose the correct option:

$\text{A}\text{. log}{\displaystyle \frac{\pi }{2}}$
$\text{B}\text{. logi}{\displaystyle \frac{\pi }{2}}$
$\text{C}\text{. log}{\displaystyle \frac{\pi }{2}}+{\displaystyle \frac{i\pi }{2}}$
$\text{D}\text{. log}{\displaystyle \frac{\pi }{2}}-{\displaystyle \frac{i\pi }{2}}$

Answer 1

Hint – We know, $z=e^{i\theta }=\cos \theta +i\sin \theta$, where z is a complex number. Now, if there is no real part in a complex number then,
$$\begin{gathered} \cos \theta =0 \\ \Rightarrow \theta ={\displaystyle \frac{\pi }{2}} \end{gathered}$$
Hence, we can say, if
$$\begin{gathered} z=i \\ \Rightarrow i=e^{i{\displaystyle \frac{\pi }{2}}} \end{gathered}$$
Use this to solve.

Complete step by step answer:
We have been asked to find $\log (\log i)$.
So, using the hint we can write, $i=e^{i{\displaystyle \frac{\pi }{2}}}$.
So, the given equation $\log (\log i)$ will transform into-
$\log (\log e^{i{\displaystyle \frac{\pi }{2}}})$.
Now, solving it further, we get-
$$\begin{gathered} \log (\log e^{i{\displaystyle \frac{\pi }{2}}})=\log \left(i.{\displaystyle \frac{\pi }{2}}\right) \\ =\log \left({\displaystyle \frac{i\pi }{2}}\right) \end{gathered}$$
Hence, the value of $\log (\log i)=\log \left({\displaystyle \frac{i\pi }{2}}\right)$.
Therefore, the correct option is B.

Note – Whenever solving such types of questions, always use the concepts of complex numbers to solve the question step by step. As mentioned in the solution, let z = I, since it does not have a real part so keep the $\cos \theta =0$, from here we can find the value of theta as 90 degrees, and then our equation will be easier to solve.