Question

Assertion :If $a=y^2,b=z^2,c=x^2,$ then $8{\log }_b{x}^3.{\log }_a{y}^3.{\log }_c{z}^3=27$ Reason: ${\log }_{b}a.{\log }_{c}b={\log }_{c}a$ and also ${\log }_{b}a=\frac{1}{{\log }_{a}b}$

• 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 correct but Reason is incorrect.
• D. Both Assertion and Reason are incorrect.

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.

As ${\log }_{b}a.{\log }_{c}b={\log }_{c}a\Rightarrow \frac{\log a}{\log b}.\frac{\log b}{\log c}=\frac{\log a}{\log c}={\log }_{c}a$
Now,
$8{\log }_b{x}^3\times {\log }_a{y}^3\times {\log }_c{z}^3=8\times 3\times 3\times 3\times {\log }_{b}x\times {\log }_{a}y\times {\log }_{c}z=216\times {\log }_{b}x\times {\log }_{a}y\times {\log }_{c}z$
$=216\times {\log }_b{c}^{\frac{1}{2}}\times {\log }_a{a}^{\frac{1}{2}}\times {\log }_c{b}^{\frac{1}{2}}$
$=\frac{216}{8}{\log }_{b}c\times {\log }_{a}a\times {\log }_{c}b$
$=27\frac{\log c}{\log b}\times \frac{\log a}{\log a}\frac{\log b}{\log c}=27$