Question

Solve the equation
$(a^4-2a^2{b}^2+b^4{)}^{x-1}=(a-b{)}^{2x}(a+b{)}^{-2}$.

Answer 1

${(a^4-2a^2{b}^2+b^4)}^{x-1}=(a-b{)}^{2x}(a+b{)}^{-2}$

Applying $\log$ on both the sides gives

$(x-1)\log (a^4-2a^2{b}^2+b^4)=2x\log (a-b)-2\log (a+b)\Rightarrow (x-1)\log {(a^2-b^2)}^2=2x\log (a-b)-2\log (a+b)\Rightarrow 2(x-1)\log (a^2-b^2)=2x\log (a-b)-2\log (a+b)\Rightarrow 2(x-1)\left(\log \right(a+b)+\log (a-b\left)\right)=2x\log (a-b)-2\log (a+b)\Rightarrow x\log (a+b)=\log (a-b)\Rightarrow x=\frac{\log (a-b)}{\log (a+b)}$