Question

Assertion :The equation ${\log }_{\frac{1}{2+\left|x\right|}}(5+x^2)={\log }_{(3+x^2)}(15+\sqrt{x})$ has real solutions. Reason: ${\log }_{\frac{1}{b}}a=-{\log }_{b}a$ (where $a.b>0$ and $b$ $\ne$1) and if number and base both are greater than unity then the number is positive.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is incorrect but Reason is correct
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is incorrect but Reason is correct

Reason:
Consider, ${\log }_{\frac{1}{b}}a$
$=\frac{{\log }_{e}a}{{\log }_e\left(\frac{1}{b}\right)}$
$=\frac{{\log }_{e}a}{-{\log }_{e}b}$
$=-{\log }_{b}a$
Hence, reason is true.
${\log }_{\frac{1}{2+\left|x\right|}}(5+x^2)={\log }_{(3+x^2)}(15+\sqrt{x})$
$\Rightarrow -{\log }_{2+\left|x\right|}(5+x^2)={\log }_{(3+x^2)}(15+\sqrt{x})$
Here, $LHS<0$ and $RHS>0$
Hence, there is no solution for the given equation.
Hence, assertion is incorrect and reason is correct.