Question

log (logi)= please find this problem​

Answer 1

Given:

Given expression is: $\text{log} (\text{log}\ i)$

To find:

We should find the value of given expression.

Solution:

We can find the value of given expression by simplifying it.

Consider the given expression,

$\Rightarrow \text{log} (\text{log}\ i)$.....(i)

We know that,

$\Rightarrow i=e^{i\theta}$

$\Rightarrow i=cos\theta+i\ sin\theta$

Let, $\theta=\frac{\pi}{2}$

Now substitute it in above expression,

$\Rightarrow i=0+i$

Hence, the value will be $i=e^{i\frac{\pi}{2}}$...(ii)

Substitute equation (i) in equation (ii), we get

$\Rightarrow \text{log} (\text{log}\ i)$

$\Rightarrow \text{log} (\text{log}\ e^{i\frac{\pi}{2}})$

We know that, $m \ \text{log}\ a=\text{log}\ a^m$.

Applying the above formula, we get

$\Rightarrow \text{log}( i\frac{\pi}{2}(\text{log}\ e))$

We know that, the value of $\text{log} \ e=1$.

$\Rightarrow \text{log}( i\frac{\pi}{2}(1))$

$\Rightarrow \text{log}( i\frac{\pi}{2})$

We know that, $\text{log} \ ab=\text{log}\ a+\text{log}\ b$.

Applying the above formula, we get

$\Rightarrow \text{log}\ i}+\text{log}\ \frac{\pi}{2}$

$\Rightarrow \text{log}\ e^{i\frac{\pi}{2}}+\text{log}\ \frac{\pi}{2}$

Simplifying the above expression,

$\Rightarrow{i\frac{\pi}{2}}+\text{log}\ \frac{\pi}{2}$

Therefore, the value of given expression is equal to ${i\frac{\pi}{2}}+\text{log}\ \frac{\pi}{2}$.