Question

For $x,a>0$ the root(s) of the equation ${\log }_{ax}a+{\log }_x{a}^2+{\log }_{a^{2}x}{a}^3=0$ is (are) given by

• A. $a^{-4/3}$
• B. $a^{-3/4}$
• C. $a^{-1/2}$
• D. $a^{-2/3}$

Answer 1

Answer: (A), (C)

The correct options are
A $a^{-4/3}$
C $a^{-1/2}$

$\frac{\log a}{\log a+\log x}+\frac{2\log a}{\log x}+\frac{3\log a}{2\log x+\log a}=0$

$⟹\frac{1}{A+X}+\frac{2}{X}+\frac{3}{2A+X}=0$

$⟹\frac{-1}{A+X}=\frac{2}{X}+\frac{3}{2A+X}$

$⟹X=\frac{-4A}{3}\frac{-A}{2}$

$⟹x=a^{\frac{-4}{3}},a^{\frac{-1}{2}}$