Solve-for-x-in-i-x-2-

Question Number 216485 by Tawa11 last updated on 08/Feb/25

Solve for x in : i^{x} = 2

Answered by issac last updated on 09/Feb/25

x \cdot \ln (i) = \ln (2)

\ln (i) = \frac{π}{2} i

∴ x = - \frac{2}{π} i \cdot \ln (2)

Commented by Tawa11 last updated on 09/Feb/25

Thanks sir

Answered by Ghisom last updated on 09/Feb/25

x \in C

x = a + b i, a, b \in R

i = e^{i \frac{π}{2}}

i^{x} = e^{i \frac{π}{2} (a + b i)} = e^{- \frac{b π}{2} + i \frac{a π}{2}} = \frac{1}{e^{\frac{b π}{2}}} e^{i \frac{a π}{2}} = 2

\Rightarrow \frac{1}{e^{\frac{b π}{2}}} = 2 \land \frac{a π}{2} = 2 n π

\Rightarrow b = - \frac{2 \ln 2}{π} \land a = 4 n

x = 4 n - \frac{2 \ln 2}{π} i; n \in Z

Commented by Tawa11 last updated on 09/Feb/25

Thanks sir

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