Question

The value of $x^{\ln y-\ln z}\cdot y^{\ln z-\ln x}\cdot z^{\ln x-\ln y}$ is

Answer 1

let $t=x^{lny-lnz}\cdot y^{lnz-lnx}\cdot z^{lnx-lny}$
Taking logarithm on both sides, we get
$\ln t=(\ln y-\ln z)\ln x+(\ln z-\ln x)\ln y+(\ln x-\ln y)\ln z$
$\Rightarrow \ln t=\ln y\ln x-\ln z\ln x+\ln z\ln y-\ln x\ln y+\ln x\ln z-\ln y\ln z=0$
$\Rightarrow t=1$
Ans: 1