Question

If $x , y ( x \neq 1)$ 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 $\left.a , b \in N \right)$. Then find the smallest value of $|a-b|$.

Answer 1

$2^{\ln x}=3^{\ln y}$ ....(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 $