To find the value, let log_{2\sqrt{3}} (1728) = x. This implies (2\sqrt{3})^x = 1728. Squaring both sides or expressing 1728 in terms of prime factors: 1728 = 12^3. Now, (2\sqrt{3})^2 = 4 * 3 = 12. Therefore, 1728 = ( (2\sqrt{3})^2 )^3 = (2\sqrt{3})^6. Comparing the powers, x = 6. The other options are incorrect because they do not satisfy the exponential form of the logarithmic equation.