Question

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

Answer 1

$2^x=3^y={12}^{z}Let{2}^x=3^y={12}^z=k\Rightarrow 2=k^{\frac{1}{x}},3=k^{\frac{1}{y}},12=k^{\frac{1}{z}}Now,12=k^{\frac{1}{z}}\Rightarrow 2^2\times 3=k^{\frac{1}{z}}\Rightarrow (k^{\frac{1}{x}}{)}^2\times (k^{\frac{1}{y}})=k^{\frac{1}{z}}\Rightarrow k^{\frac{2}{x}}.k^{\frac{1}{y}}=k^{\frac{1}{z}}\Rightarrow k^{\frac{2}{x}+\frac{1}{y}}=k^{\frac{1}{z}}Comparingbothsides,\therefore \frac{1}{z}=\frac{1}{y}+\frac{2}{x}HenceProved.$