help8

log(384/5)+log(81/32)+3log(5/3)+log(1/9)=?

\(\begin{array}{rcll} \log{ \left( \frac{384}{5} \right) } + \log{ \left( \frac{81}{32} \right) } +3 \cdot \log{ \left( \frac{5}{3} \right) } + \log{ \left( \frac{1}{9} \right) } &=&\\ &=& \log{ \left( \frac {384}{5}\cdot \frac {81}{32} \cdot \frac {5^3}{3^3} \cdot \frac{1}{9} \right) } \\ &=& \log{ \left( \frac {3\cdot 2^7}{5}\cdot \frac {3^4}{2^5} \cdot \frac {5^3}{3^3} \cdot \frac{1}{3^2} \right) } \\ &=& \log{ \left( \frac{2^7}{2^5} \cdot \frac{3\cdot 3^4 }{3^3\cdot 3^2} \cdot \frac{5^3}{5} \right) } \\ &=& \log{ \left( \frac{2^7}{2^5} \cdot \frac{ 3^5 }{3^5} \cdot \frac{5^3}{5} \right) } \\ &=& \log{ \left( \frac{2^7}{2^5} \cdot \frac{5^3}{5} \right) } \\ &=& \log{ ( 2^2 \cdot 5^2 ) } \\ &=& \log{ ( (2\cdot 5 )^2 ) } \\ &=& \log{ ( 10^2 ) } \\ &=& 2\cdot \log{ ( 10 ) } \qquad & | \qquad \log{ ( 10 ) } = 1\\ &=& 2\cdot 1\\ \log{ \left( \frac{384}{5} \right) } + \log{ \left( \frac{81}{32} \right) } +3 \cdot \log{ \left( \frac{5}{3} \right) } + \log{ \left( \frac{1}{9} \right) } &=& 2 \end{array}\)