Question

If $\log x: \log y: \log z=(y-z):(z-x):(x-y)$, then

• A. $x^x y^y z^z=1$
• B. $x^y y^z z^x=1 $
• C. $ x y z=1$
• D. $x^{1 / x} y^{1 / y} z^{1 / z}=1$

Answer 1

Answer: (A), (C)

$\frac{\log x}{y-z}=\frac{\log y}{z-x}=\frac{\log z}{x-y}=k$
$\therefore \log x=k(y-z): \log y=k(z-x): \log z=k(x-y) $
$\therefore \log x+\log y+\log z=0$
$\Rightarrow x y z=1 $
$\text { and } x \log x+y \log y+z \log z=0$
$\Rightarrow x^x y^y z^z=1$