Question

If $2x^{{\log }_{4}3}+3^{{\log }_{4}x}=27$, then $x$ is equal to

• A. $2$
• B. $4$
• C. $8$
• D. $16$

Answer 1

The correct option is D $16$

Let, $y=x^{{\log }_{4}3}$
$\Rightarrow \log y={\log }_{4}3\log x=\log x\left(\frac{\log 3}{\log 4}\right),[\because {\log }_{a}b=\frac{\log b}{\log a}]$
$\Rightarrow {\log }_{3}y={\log }_{4}x$
$\Rightarrow y=3^{{\log }_{4}x}$
Substitute $y$ in equation $2x^{{\log }_{4}3}+3^{{\log }_{4}x}=27$ we get,
$3\times 3^{{\log }_{4}x}=27$
$\Rightarrow 3^{1+{\log }_{4}x}=3^3$
Equating the power of 3
$\Rightarrow 1+{\log }_{4}x=3\Rightarrow {\log }_{4}x=2$
$\Rightarrow x=4^2=16$
Ans: D