Question

If $x={\log }_{24}12,y={\log }_{36}24$ and $z={\log }_{144}36$, then the value of $1+2xyz$ is equal to

• A. $2$
• B. $3$
• C. $1$
• D. $5$

Answer 1

The correct option is A $2$

$x={\log }_{24}12,y={\log }_{36}24,z={\log }_{144}36$
$xyz={\log }_{24}12.{\log }_{36}24.{\log }_{144}36$

$\Rightarrow xyz=\left(\frac{\log 12}{\log 24}\right)\left(\frac{\log 24}{\log 36}\right)\left(\frac{\log 36}{\log {12}^2}\right)[\because {\log }_{a}b=\frac{\log b}{\log a}]$

$\Rightarrow xyz=\frac{\log 12}{\log {12}^2}=\frac{1}{2}[\because \log a^m=m\log a]$

$\Rightarrow 1+2xyz=2$
Ans: A