Question

If logx/y-z=logy/z-x=logz/x-y then prove that xyz=1

Answer 1

Let,

$\dfrac{\log{x}}{y-z}=\dfrac{\log{y}}{z-x}=\dfrac{\log{z}}{x-y}=k$

Then,

$\dfrac{\log{x}}{y-z}=k\quad\implies\quad\log{x}=k(y-z)\quad\implies\quad x=10^{k(y-z)}\\\\\\\dfrac{\log{y}}{z-x}=k\quad\implies\quad\log{y}=k(z-x)\quad\implies\quad y=10^{k(z-x)}\\\\\\\dfrac{\log{z}}{x-y}=k\quad\implies\quad\log{z}=k(x-y)\quad\implies\quad z=10^{k(x-y)}$

Now,

$xyz=10^{k(y-z)}\cdot10^{k(z-x)}\cdot10^{k(x-y)}\\\\\\xyz=10^{ky-kz}\cdot10^{kz-kx}\cdot10^{kx-ky)}\\\\\\xyz=10^{ky-kz+kz-kx+kx-ky}\\\\\\xyz=10^0\\\\\\xyz=1$

Hence Proved!