Question

Solve the following equation:
${\log }^2\left(100x\right)-{\log }^2\left(10x\right)+{\log }^{2}x=6$

Answer 1

${\log }^2\left(100x\right)-{\log }^2\left(10x\right)+{log}^{2}x=6\Rightarrow (2+\log x{)}^2-(1+{\log x)}^2+(\log x{)}^2=6\Rightarrow 3+2\log x+{\left(\log x\right)}^2=6\Rightarrow {\left(\log x\right)}^2+2\log x-3=0\Rightarrow \log x=\frac{-2\pm \sqrt{16}}{2}\Rightarrow \log x=1,-3.\Rightarrow x=10,{10}^{-3}$