Question

Solve the following equation:
${\log }_{5}x+{\log }_{25}x={\log }_{1/5}\sqrt{3}$

Answer 1

${\log }_{5}x+{\log }_{25}x={\log }_{\frac{1}{5}}\sqrt{3}\Rightarrow {\log }_{5}x+\frac{1}{2}{\log }_{5}x=\frac{-1}{2}{\log }_{5}3\Rightarrow {\log }_{5}x{\left(3x\right)}^{\frac{1}{2}}=0\Rightarrow {x\left(3x\right)}^{\frac{1}{2}}=1\Rightarrow {3x}^3=1\Rightarrow x={\left(\frac{1}{3}\right)}^{\frac{1}{3}}\Rightarrow x=\frac{1}{\sqrt[3]{33}}.$