help 8.4

log(75÷16)-2log(5÷9)+log(32÷243)=log(2)

$\begin{array}{rcll} \log{( \frac{75}{16} )}-2\cdot \log{(\frac{5}{9} )}+ \log{( \frac{32}{243} )} &\overset{?}{=}& \log{(2)} \\ \log{( \frac{75}{16} )}- \log{(\frac{5^2}{9^2} )}+ \log{( \frac{32}{243} )} &\overset{?}{=}& \log{(2)} \\ \log{( \frac{75}{16}\cdot \frac{1}{\frac{5^2}{9^2} } \cdot \frac{32}{243} )} &\overset{?}{=}& \log{(2)} \\ \log{( \frac{75}{16}\cdot \frac{9^2}{ 5^2 } \cdot \frac{32}{243} )} &\overset{?}{=}& \log{(2)} \\ \log{( \frac{ 75\cdot 9^2 \cdot 32 } {16\cdot 5^2\cdot 243} ) } &\overset{?}{=}& \log{(2)} \\ \log{( \frac{ 75\cdot 81 \cdot 32 } {16\cdot 25\cdot 243} ) } &\overset{?}{=}& \log{(2)} \\ \log{( \frac{75}{25} \cdot \frac{32}{16}\cdot \frac{81}{243} ) } &\overset{?}{=}& \log{(2)} \\ \log{( \frac{3}{1} \cdot \frac{2}{1}\cdot \frac{1}{3} ) } &\overset{?}{=}& \log{(2)} \\ \log{( 2 ) } &=& \log{(2)} \\ \end{array}$