1. \( \begin{array}{l}\text { Simplify (without using a } \\ \text { calculator): } \\ \frac{\log 6-\log 2}{\log 9(2 \log 5+\log 4)}\end{array} \)
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To simplify \( \frac{\log 6 - \log 2}{\log 9 (2 \log 5 + \log 4)} \), we can first use the properties of logarithms. The difference of logs gives: \[ \log 6 - \log 2 = \log \left(\frac{6}{2}\right) = \log 3. \] Next, for the denominator, we can combine \(2 \log 5 + \log 4\) as follows: \[ 2 \log 5 + \log 4 = \log(5^2) + \log(4) = \log(25) + \log(4) = \log(25 \cdot 4) = \log(100). \] Since \(100 = 10^2\), we have: \[ \log 100 = \log(10^2) = 2 \log 10 = 2. \] Now, the denominator can be simplified: \[ \log 9 (2 \log 5 + \log 4) = \log 9 \cdot \log 100. \] We know \( \log 9 = \log(3^2) = 2 \log 3 \). Therefore: \[ \log 9 \cdot \log 100 = (2 \log 3) \cdot 2 = 4 \log 3. \] Now our entire expression looks like this: \[ \frac{\log 3}{4 \log 3}. \] Now, the \(\log 3\) terms in the numerator and denominator cancel out (provided \(\log 3 \neq 0\)), leaving: \[ \frac{1}{4}. \] Thus, the simplified result is: \[ \boxed{\frac{1}{4}}. \]
