Question

Range of real values of ${}^{\prime }{p}^{\prime }$ for which the equation $2{\log }_3^{2}x-\left|{\log }_{3}x\right|+p=0$ has 4 solutions is

• A. $(0,\frac{1}{8})$
• B. $(0,\frac{1}{8}]$
• C. $[0,\frac{1}{8})$
• D. $[0,\frac{1}{8}]$

Answer 1

The correct option is A $(0,\frac{1}{8})$

Given $2{\log }_3^{2}x-\left|{\log }_{3}x\right|+p=0$

this is a quadratic equation in ${\log }_{3}x$

${\log }_{3}x=\frac{-1\pm \sqrt{1^2-8p}}{4}=\frac{-1\pm \sqrt{1-8p}}{4}$

For real solution

$D\ge 01-8p\ge 01\ge 8pp\le \frac{1}{8}$

For four solutions, $p\ne 0$ and $p\ne \frac{1}{8}$ as on these values we get only two solutions.

Hence, $p\in (0,\frac{1}{8})$