Logarithmic Simplification

Note that $(7 - 4\sqrt{3})^2 = 49 + 48 - 56\sqrt{3} = 97 - 56\sqrt{3}$. So $97 - 56\sqrt{3} = (7 - 4\sqrt{3})^2$. Also $7 - 4\sqrt{3} = \frac{1}{7 + 4\sqrt{3}} = (7 + 4\sqrt{3})^{-1}$. Thus $97 - 56\sqrt{3} = (7 + 4\sqrt{3})^{-2}$. Denominator: $\log\sqrt{7 + 4\sqrt{3}} = \frac{1}{2} \log(7 + 4\sqrt{3})$. Numerator: $\log((7 + 4\sqrt{3})^{-2}) = -2 \log(7 + 4\sqrt{3})$. Ratio = $\frac{-2 \log(7 + 4\sqrt{3})}{\frac{1}{2} \log(7 + 4\sqrt{3})} = -4$.