If $(1 + i)^{i}$ = A+iB. Find the value of ${l o g}_{e}$ (A+iB)

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Solution

The correct option is B

A+iB = $(1 + i)^{i}$

Taking ${l o g}_{e}$ on both sides

${l o g}_{e}$ (A+iB) = ${l o g}_{e}$ $(1 + i)^{i}$

= i ${l o g}_{e} (1 + i)$

= i ${l o g}_{e} (\sqrt{2} (c o s \frac{π}{4} + i s i n \frac{π}{4}))$

= i ${l o g}_{e} (\sqrt{2} e^{i \frac{π}{4}})$

= i $[{l o g}_{e} \sqrt{2} + {l o g}_{e} e^{i} \frac{π}{4}] = i {l o g}_{e} \sqrt{2} + i^{2} \frac{π}{4} = i {l o g}_{e} \sqrt{2} - \frac{π}{4}$

= $\frac{i}{2} {l o g}_{e} 2 - \frac{π}{4}$

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