Question

The value of $x$ which satisfies the equation $\log 4+(1+\frac{1}{2x})\log 3=\log (\sqrt[x]{x3}+27)$ is

• A. $x\in \varphi$
• B. $1$
• C. $4$
• D. none of these

Answer 1

The correct option is D none of these

$\log 4+(1+\frac{1}{2x})\log 3=\log (\sqrt[x]{x3}+27)$
$\Rightarrow \log 4+\log 3+\log 3^{\frac{1}{2x}}=\log (3^{\frac{1}{x}}+27)$
$\Rightarrow \log \left({12.3}^{\frac{1}{2x}}\right)=\log (3^{\frac{1}{x}}+27)[\because \log a+\log b+\log c=\log (abc\left)\right]$
$\Rightarrow {12.3}^{\frac{1}{2x}}=3^{\frac{1}{x}}+27$
Let $3^{\frac{1}{2x}}=y$
$\Rightarrow y^2-12y+27=0\Rightarrow y=9$ or $y=3$
$\Rightarrow x=\frac{1}{4}$ and $x=\frac{1}{2}$