Expand each logarithm. 1) \( \log (6 \cdot 11) \) 2) \( \log (5 \cdot 3) \) 3) \( \log \left(\frac{6}{11}\right)^{5} \) 4) \( \log \left(3 \cdot 2^{3}\right) \) 5) \( \log \frac{2^{4}}{5} \) 6) \( \log \left(\frac{6}{5}\right)^{6} \) 7) \( \log \frac{x}{y^{6}} \) 8) \( \log (a \cdot b)^{2} \) 9) \( \log \frac{u^{4}}{v} \) 10) \( \log \frac{x}{y^{5}} \) 11) \( \log \sqrt[3]{x \cdot y \cdot z} \) 12) \( \log \left(x \cdot y \cdot z^{2}\right) \)
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1) \( \log (6 \cdot 11) = \log 6 + \log 11 \) 2) \( \log (5 \cdot 3) = \log 5 + \log 3 \) 3) \( \log \left(\frac{6}{11}\right)^{5} = 5(\log 6 - \log 11) \) 4) \( \log \left(3 \cdot 2^{3}\right) = \log 3 + \log (2^{3}) = \log 3 + 3 \log 2 \) 5) \( \log \frac{2^{4}}{5} = \log (2^{4}) - \log 5 = 4 \log 2 - \log 5 \) 6) \( \log \left(\frac{6}{5}\right)^{6} = 6(\log 6 - \log 5) \) 7) \( \log \frac{x}{y^{6}} = \log x - 6 \log y \) 8) \( \log (a \cdot b)^{2} = 2(\log a + \log b) = 2 \log a + 2 \log b \) 9) \( \log \frac{u^{4}}{v} = \log (u^{4}) - \log v = 4 \log u - \log v \) 10) \( \log \frac{x}{y^{5}} = \log x - 5 \log y \) 11) \( \log \sqrt[3]{x \cdot y \cdot z} = \frac{1}{3}(\log (x \cdot y \cdot z)) = \frac{1}{3}(\log x + \log y + \log z) \) 12) \( \log \left(x \cdot y \cdot z^{2}\right) = \log x + \log y + \log (z^{2}) = \log x + \log y + 2 \log z \)
