What is X when raised to the 3-5i power and the answer is 23-14i?

Answer 1

X = 1.31356+0.612045*i

Steps to solve, take the natural log of both sides:

ln(X^(3-5i)) = ln(23-14i).

(3-5i)*ln(X) = ln(23-14i). Convert 23-14i to exponential form: A*e^(iΘ) {A = 26.926 and Θ = -0.54679 radians}

(3-5i)*ln(X) = ln(A*e^(iΘ))= ln(A) + iΘ = ln(26.926) - 0.54679i.

divide by (3-5i): ln(X) = (ln(A) + iΘ) / (3-5i) = (3.2931 - 0.54679i)/(3-5i)

So we have ln(X) = 0.370978 + 0.436033i, then:

e^(ln(X)) = e^(0.370978 + 0.436033i) --> X = 1.31356+0.612045*i