Question

If $2^x=3^y={12}^z$, show that $\frac{1}{z}=\frac{1}{y}+\frac{2}{x}$

Answer 1

$2^x=3^y={12}^z=k\left(let\right)$

$x={\log }_{2}ky={\log }_{3}kz={\log }_{12}k$

$\frac{1}{z}=\frac{1}{y}+\frac{2}{x}=\frac{1}{{\log }_{3}k}+\frac{2}{{\log }_{2}k}={\log }_{k}3+2{\log }_{k}2={\log }_{k}3+{\log }_{k}4$

${\log }_{k}12=\frac{1}{z}$