Question

If $x,y(x\ne 1)$ satisfy both the equation $2^{\ln x}=3^{\ln y}$ and $\left({\ln }^{2}x\right)=\left({\ln }^{3}y\right)$. The value of the expression $\sqrt{\ln y}+\sqrt[3]{\ln x}$ can be expressed as ${\log }_{b}a$ (where $a,b\in N)$. Then find the smallest value of $|a-b|$.

Answer 1

Answer: 7

$2^{\ln x}=3^{\ln y}$ ....(1)
Taking $\ln$ both the sides
$\ln x\cdot \ln 2=\ln y\cdot \ln 3\Rightarrow \ln x={\log }_{2}3\cdot \ln y$
Put in eqaution
${\ln }^{2}x={\ln }^{3}y\Rightarrow \ln y={\log }_2^{2}3\Rightarrow \ln x={\log }_2^{3}3\Rightarrow {\log }_{2}3+{\log }_{2}3={\log }_{2}9$
$\Rightarrow a=9,b=2$
$|a-b|=|9-2|=7$