If ${2}^{x}={3}^{y}={6}^{-z}$ and $x,y,z\ne 0$ then prove that:

$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$

I have tried starting with taking logartithms, but that gives just some more equations.

Any specific way to solve these type of problems?

$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$

I have tried starting with taking logartithms, but that gives just some more equations.

Any specific way to solve these type of problems?