Question

Solve the following equation:
$x^{1-\frac{1}{3}\log x^2}=\frac{1}{\sqrt[3]{3100}}$

Answer 1

$x^{1-\frac{1}{3}{logx}^2}=\frac{1}{\sqrt[3]{3100}}\Rightarrow 1-\frac{1}{3}{logx}^2={log}_x{10}^{\frac{-2}{3}}\Rightarrow 1-\frac{2}{3}logx=\frac{-2}{3}{log}_{x}10$

Let ${log}_{10}x=a.$

$1-\frac{2a}{3}=\frac{-2}{3a}\Rightarrow 3a-{2a}^2=-2{\Rightarrow 2a}^2-3a-2=0\Rightarrow {2a}^2-4a+a-2=0\Rightarrow (2a+1)(a-2)=0$

$\Rightarrow a=\frac{-1}{2}$ or $a=2$

$\Rightarrow lo{g}_{10}x=\frac{-1}{2}$ or ${log}_{10}x=2$

$\Rightarrow x=\frac{1}{\sqrt{100}}$ or $100.$