Question

Find all integral solution of the equation, $4{\log }_{\frac{x}{2}}\left(\sqrt{x}\right)+2{\log }_{4x}\left(x^2\right)=3{\log }_{2x}\left(x^3\right)$.

Answer 1

$4{\log }_{x/2}\sqrt{x}+2{\log }_{4x}{x}^2=3{\log }_{2x}\left(x^3\right)$

$\Rightarrow$ $4\frac{\log \sqrt{x}}{\log \frac{x}{2}}+2\frac{\log x^2}{\log 4x}=3\frac{\log x^3}{\log 2x}$

$\Rightarrow$ $2\frac{\log x}{\log \frac{x}{2}}+4\frac{\log x}{\log 4x}=9\frac{\log x}{\log 2x}$

$\Rightarrow$ $\log x\left[\frac{2\log 4x+4\log \frac{x}{2}}{\log \frac{x}{2}\cdot \log 4x}\right]=9\frac{\log x}{\log 2x}$

$\Rightarrow$ $[2(\log 4+\log x)+4(\log x-\log 2)]\log 2x=9\log \left(\frac{x}{2}\right)\cdot \log 4x$

$\Rightarrow$ $[2[2\log 2+\log x]+4\log x-4\log 2](\log 2+\log x)=9\log \frac{x}{2}\cdot \log 4x$
$\Rightarrow$ $6\log x[\log 2+\log x]=9(\log x-\log 2)(2\log 2+\log x)$
$\Rightarrow$ $6\log x[\log 2+\log x]=9[{\left(\log x\right)}^2-{\left(\log 2\right)}^2+\log 2\cdot \log x-{\left(\log 2\right)}^2]$
$\Rightarrow$ $3{\left(\log x\right)}^2+\log x\left(3\log 2\right)-18{\left(\log 2\right)}^2=0$
$\log x=\frac{-3\log 2\pm \sqrt{9{\left(\log 2\right)}^2+216{\left(\log 2\right)}^2}}{6}$
$\log x=\frac{-3\log 2\pm 15\log 2}{6}$
$\log x=2\log 2,-3\log 2$
$\log x=\log 4,\log \frac{1}{8}$
$x=4,\frac{1}{8}$