Question

The number of real root(s) of the equation ${\log }_{\sqrt[3]{3x}}x+{\log }_{3x}\sqrt{x}=0$ is/are

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is A $1$

${\log }_{\sqrt[3]{3x}}x+{\log }_{3x}\sqrt{x}=0$

$\Rightarrow 3{\log }_{x}x+\frac{1}{2}{\log }_{3x}x=0[\because {\log }_{a^n}{b}^m=\frac{m}{n}{\log }_{a}b]$

$\Rightarrow {\log }_{3x}x=-6[\because {\log }_{a}a=1]$

$\Rightarrow x=(3x{)}^{-6}$

$\Rightarrow x^7=3^{-6}$

$\Rightarrow x=3^{-\frac{6}{7}}$

$\Rightarrow$ one real solution.