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

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To condense the expression $$ 2(\log 18 - \log 3) + \frac{1}{2} \log \frac{1}{16} $$ into a single logarithm, follow these steps:

1. **Apply the Quotient Rule:**
   $$
   \log 18 - \log 3 = \log \left( \frac{18}{3} \right) = \log 6
   $$
   So, the expression becomes:
   $$
   2 \log 6 + \frac{1}{2} \log \frac{1}{16}
   $$

2. **Apply the Power Rule:**
   $$
   2 \log 6 = \log 6^2 = \log 36
   $$
   $$
   \frac{1}{2} \log \frac{1}{16} = \log \left( \frac{1}{16} \right)^{\frac{1}{2}} = \log \frac{1}{4}
   $$
   Now, the expression is:
   $$
   \log 36 + \log \frac{1}{4}
   $$

3. **Combine Using the Product Rule:**
   $$
   \log 36 + \log \frac{1}{4} = \log \left( 36 \times \frac{1}{4} \right) = \log 9
   $$

So, the condensed form of the expression is:

$$
\log 9
$$

**Answer:** 
$$\log 9$$
$$\log 9$$

Condense the expression.

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