Question

Solve the following equation:
${\log }_4+(1+\frac{1}{2x})\log 3=\log (\sqrt[x]{x3}+27)$

Answer 1

$\log 4+(1+\frac{1}{2x})\log 3=\log (\sqrt[x]{x3}+27)\Rightarrow \log 4+\log \left[3^{(1+\frac{1}{2x})}\right]=\log (\sqrt[x]{x3}+27)1$

Let $3^{\frac{1}{2x}}=a\log 4+\log \left(3a\right)=\log (a^2+27)\Rightarrow 12a=a^2+27\Rightarrow a^2-12a+27=0a=9,3.$

Therefore, $3^{\frac{1}{2x}}=9,3\Rightarrow \frac{1}{2x}=2,1\Rightarrow x=\frac{1}{4},\frac{1}{2}$