Question

If $\frac{logx}{y-z}=\frac{logy}{z-x}=\frac{logz}{x-y}$ , then $xyz=?$

Answer 1

Let $\frac{\log x}{y-z}=\frac{\log y}{z-x}=\frac{\log z}{x-y}=k$ (say)

$\therefore \log x=k(y-z)$, $\log y=k(z-x)$, $\log z=k(x-y)$

$Now,\log \left(xyz\right)=\log x+\log y+\log z$

$\therefore \log \left(xyz\right)=k(y-z+z-x+x-y)$

$\therefore \log \left(xyz\right)=0$

$\therefore \log \left(xyz\right)=\log 1$

$\therefore xyz=1$