Find the numerical value of the log expression. \[ \begin{array}{c}\log a=-8 \quad \log b=-9 \quad \log c=-9 \\ \log \frac{a^{2}}{b^{5} c^{8}}\end{array} \]

To solve the logarithmic expression \(\log \frac{a^{2}}{b^{5} c^{8}}\), we can use the properties of logarithms. First, we know from the properties of logarithms: \[ \log \frac{X}{Y} = \log X - \log Y \] and \[ \log (X^n) = n \cdot \log X \] Using the first property, we can rewrite the original expression as follows: \[ \log \frac{a^{2}}{b^{5} c^{8}} = \log a^{2} - \log (b^{5} c^{8}) \] Next, we apply the second property on the terms inside the logarithm: \[ \log a^{2} = 2 \log a \] and \[ \log (b^{5} c^{8}) = \log b^{5} + \log c^{8} = 5 \log b + 8 \log c \] So now, substituting these into our expression, we get: \[ \log \frac{a^{2}}{b^{5} c^{8}} = 2 \log a - (5 \log b + 8 \log c) \] Now substituting the given values: \(\log a = -8\), \(\log b = -9\), and \(\log c = -9\): \[ = 2(-8) - (5(-9) + 8(-9)) \] Calculating \(2(-8)\): \[ = -16 \] Now calculating \(5(-9) + 8(-9)\): \[ = -45 - 72 = -117 \] Putting it all together, we have: \[ -16 - (-117) = -16 + 117 = 101 \] Thus, the final value of the expression is: \[ \log \frac{a^{2}}{b^{5} c^{8}} = \boxed{101} \]