Question

If 2^x=3^y=12^z ,prove that X=2yz/y-z​

Answer 1

Answer:

$2^{x} = 3^{y} = 12^{z} = k$

=> $2^{x} = k$

$2 =\sqrt[x]{k}$

$2 = k^{\frac{1}{x} }$

=> $3^{y} = k$

$3 =\sqrt[y]{k}$

$3 = k^{\frac{1}{y} }$

=> $12^{z} = k$

$12 =\sqrt[z]{k}$

$12 = k^{\frac{1}{z} }$

and we know that 12 = $2^{2} * 3$

putting the value of numbers that we found above

$k^{\frac{1}{z} } = k^{\frac{2}{x} } * k^{\frac{1}{y} }$

$k^{\frac{1}{z} } = k^{\frac{2}{x} + \frac{1}{y} }$

since the bases are same thus powers will also be same

$\frac{1}{z} = \frac{2}{x} + \frac{1}{y}$

$\frac{1}{z} - \frac{1}{y} = \frac{2}{x}$

$\frac{y - z}{yz} = \frac{2}{x}$

$x(\frac{y - z}{yz}) = 2$

$x = \frac{2yz}{y - z}$