Complex Logarithm (Part 2)

Algebra Level 3

3[ln(32+12i)ln(3212i)]=A+Bi\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

AA and BB are real numbers satisfying the equation above. Find the minimum value of A2+B2A^2 + B^2 to 2 decimal places.

Notations:

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