Question

The number of real solution(s) of the equation ${\left(x^{{\log }_{10}3}\right)}^2-3^{{\log }_{10}x}=2$ is

Answer 1

${\left(x^{{\log }_{10}3}\right)}^2-3^{{\log }_{10}x}=2$
$\Rightarrow {\left(x^{{\log }_{10}3}\right)}^2-x^{{\log }_{10}3}=2(\because a^{{\log }_{b}c}=c^{{\log }_{b}a})$
Assuming $t=x^{{\log }_{10}3}$
$\Rightarrow t^2-t-2=0$
$\Rightarrow t=-1,2$
$t=-1$ is not possible.
$\therefore t=2$
$\Rightarrow x^{{\log }_{10}3}=2$
$\Rightarrow x=2^{{\log }_{3}10}$

Hence, the number of real solution is $1$.