Complex Logarithm (Part 2)

Algebra Level 3

\[\large 3 \left[\ln \left(\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) - \ln \left(\frac{\sqrt{3}}{2} - \frac{1}{2}i\right) \right] = A +Bi\]

\(A\) and \(B\) are real numbers satisfying the equation above. Find the minimum value of \(A^2 + B^2\) to 2 decimal places.

Notations:

  • \(i = \sqrt{-1}\) denotes the imaginary unit.
  • \(\ln (\cdot)\) denotes the natural logarithm.
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