QUESTION

Find the value of $\log (\log i) =$ and choose the correct option:${\text{A}}{\text{. log}}\dfrac{\pi }{2}${\text{B}}{\text{. logi}}\dfrac{\pi }{2}${\text{C}}{\text{. log}}\dfrac{\pi }{2} + \dfrac{{i\pi }}{2}$${\text{D}}{\text{. log}}\dfrac{\pi }{2} - \dfrac{{i\pi }}{2}$

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,
$\cos \theta = 0 \\ \Rightarrow \theta = \dfrac{\pi }{2} \\$
Hence, we can say, if
$z = i \\ \Rightarrow i = {e^{i\dfrac{\pi }{2}}} \\$
Use this to solve.

We have been asked to find $\log (\log i)$.
So, using the hint we can write, $i = {e^{i\dfrac{\pi }{2}}}$.
So, the given equation $\log (\log i)$ will transform into-
$\log (\log {e^{i\dfrac{\pi }{2}}})$.
$\log (\log {e^{i\dfrac{\pi }{2}}}) = \log \left( {i.\dfrac{\pi }{2}} \right) \\ = \log \left( {\dfrac{{i\pi }}{2}} \right) \\$
Hence, the value of $\log (\log i) = \log \left( {\dfrac{{i\pi }}{2}} \right)$.
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.