Question

Solve the following equation:
${\log }_x\left(125x\right)\cdot {\log }_{25}^{2}x=1$

Answer 1

$lo{g}_x\left(125x\right).{lo{g}_{25}}^{2}x=1$

$\frac{(log125+logx)}{logx}.\frac{logx.logx}{log25.log25}=1$

$(3log5+logx)logx=4{log}^{2}5$

$letlogx=z$

$z(z+3\log 5)=4\log {}^{2}5$

$z^2+3z\log 5-4\log {}^{2}5$

$z=\frac{-3\log 5\pm \sqrt{9\log {}^{2}5+16\log {}^{2}5}}{2}$

$z=\frac{-3\log 5\pm 5\log 5}{2}$

$z=-4\log 5,z=\log 5$

$orz=\log x$

$x=5^{-4}or5$

$x=\frac{1}{625}or5$