help 8.1

log(36÷25)^3+3log(2÷9)-log(2)=2log(16÷125)

\(log(36÷25)^3+3log(2÷9)-log(2)=2log(16÷125)\\ LHS=3log(\frac{6}{5})^2+3(log2-log9)-log2\\ LHS=6log(\frac{6}{5})+3log2-3log9-log2\\ LHS=6log(6)-6log5+3log2-3log9-log2\\ LHS=6log(6)-6log5+3log2-3log3^2-log2\\ LHS=6log(6)-6log5+3log2-6log3-log2\\ LHS=6log(6)-6log5+2log2-6log3\\ LHS=2[3log(6)-3log5+log2-3log3]\\ LHS=2[log(6^3)-log5^3+log2-log3^3]\\ LHS=2[log\frac{2*6^3}{5^3*3^3}]\\ LHS=2[log\frac{2*2^3*3^3}{5^3*3^3}]\\ LHS=2[log\frac{16}{125}]\\ LHS=2log(16\div 125)\\ LHS=RHS \qquad QED\)

log(36÷25)^3+3log(2÷9)-log(2)=2log(16÷125) [prove that math]

\(\small{ \begin{array}{rcll} \log{[(\frac{36}{25} )^3]}+3\cdot \log{(\frac{2}{9} )}-\log{(2)} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{[(\frac{36^3}{25^3} )]} + \log{[(\frac{2}{9} )^3 ] }-\log{(2)} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{[(\frac{36^3}{25^3} )]} + \log{[(\frac{2^3}{9^3} ) ] }-\log{(2)} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{36^3}{25^3}\cdot \frac{2^3}{9^3} \cdot \frac12 )} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{36^3}{25^3}\cdot \frac{2^2}{9^3})} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{ (4\cdot 9)^3}{25^3}\cdot \frac{2^2}{9^3})} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{ 4^3\cdot 9^3}{25^3}\cdot \frac{2^2}{9^3})} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{ 4^3 }{25^3}\cdot \frac{2^2}{1})} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{ 4^3 }{(5^2)^3}\cdot \frac{2^2}{1})} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{ 4^3 }{(5^2)^3}\cdot 4 )} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{ 4^3 }{(5^3)^2}\cdot 4 )} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{ 4^3 }{(5^3)^2}\cdot 4 )} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{ 4^3 }{125^2}\cdot 4 )} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{ 4^4 }{125^2} )} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{ (4^2)^2 }{125^2} )} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{(\frac{ 16^2 }{125^2} )} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\ \log{ [ ( \frac{ 16 }{125} )^2 ]} & \overset{?}{=}& 2\cdot \log{(\frac{16}{125} )} \\\\ \mathbf{2\cdot \log{ ( \frac{ 16 }{125} )} } & \mathbf{=} & \mathbf{ 2\cdot \log{(\frac{16}{125} )} } \end{array} }\)