Question

Statement$-1:$ If $x^{{\log }_x(3-x{)}^2}=25$, then $x=-2$
Statement$-2:$ $a^{{\log }_{a}x}=x$, if $a>0,x>0$ and $a\ne 1$.

• A. Statement$-1$ is true, Statement$-2$ is true; Statement$-2$ is a correct explanation for Statement$-1$.
• B. Statement$-1$ is true, Statement$-2$ is true; Statement$-2$ is NOT a correct explanation for Statement$-1$.
• C. Statement$-1$ is true, Statement$-2$ is false.
• D. Statement$-1$ is false, Statement$-2$ is true.

Answer 1

The correct option is D Statement$-1$ is false, Statement$-2$ is true.

$x^{{\log }_x(3-x{)}^2}=25$
$\Rightarrow (3-x{)}^2=25,x>0\text{and}x\ne 1[\because a^{{\log }_{a}x}=x]$
$\Rightarrow 3-x=\pm 5$
$\Rightarrow x=-2,x=8$
Hence, statement$-2$ is true but statement$-1$ is false.