Question

If $x^{2{\log }_{10}x}=1000x,$ then the number of real values of $x$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

$x^{2{\log }_{10}x}=1000x$
For the $\log$ to be defined, we get
$x>0$ and $x\ne 1$
Taking ${\log }_x$ on both the sides, we get
$2{\log }_{10}x={\log }_x\left(1000x\right)$
$\Rightarrow 2{\log }_{10}x={\log }_x{10}^3+{\log }_{x}x$
$\Rightarrow 2{\log }_{10}x=3{\log }_{x}10+1$
$\Rightarrow 2{\log }_{10}x=\frac{3}{2{\log }_{10}x}+1$

Assuming ${\log }_{10}x=t$
$\Rightarrow 2t=\frac{3}{t}+1\Rightarrow 2t^2-t-3=0\Rightarrow (2t-3)(t+1)=0\Rightarrow t=-1,\frac{3}{2}\Rightarrow {\log }_{10}x=-1,\frac{3}{2}\therefore x=\frac{1}{10},{10}^{3/2}$