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

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

Last updated date: 04th Mar 2024
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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.

Complete step by step answer:
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-
$\log (\log {e^{i\dfrac{\pi }{2}}})$.
Now, solving it further, we get-
  \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 $\log (\log i) = \log \left( {\dfrac{{i\pi }}{2}} \right)$.
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.