Condense the expression. \( 2(\log 18-\log 3)+\frac{1}{2} \log \frac{1}{16} \) \( \log [?] \)

To condense the expression \( 2(\log 18 - \log 3) + \frac{1}{2} \log \frac{1}{16} \), we start by applying the properties of logarithms. 1. Using the quotient rule, we have: \[ 2(\log 18 - \log 3) = 2\log\left(\frac{18}{3}\right) = 2\log(6) \] Then apply the power rule: \[ = \log(6^2) = \log(36) \] 2. For \( \frac{1}{2} \log \frac{1}{16} \), again apply the power rule: \[ \frac{1}{2} \log \frac{1}{16} = \log \left( \left(\frac{1}{16}\right)^{1/2} \right) = \log\left(\frac{1}{4}\right) \] 3. Combine the logarithmic expressions using the product rule: \[ \log(36) + \log\left(\frac{1}{4}\right) = \log(36 \cdot \frac{1}{4}) = \log\left(\frac{36}{4}\right) = \log(9) \] So, the condensed expression is: \[ \log[9] \]