Question

If 2^x =3^y=12^z ,show that 1/x =(1/y )+(2/z)

Answer 1

$let \: {2}^{x} = {3}^{y} = {12}^{z} = k. \: \: then \\ 2 = {k}^{ \frac{1}{x} } . \: 3 = {k}^{ \frac{1}{y} } . \: 12 = {k}^{ \frac{1}{z} } \\ we \: know \: 12 = {2}^{2} \times 3 \\ {k}^{ \frac{1}{z} } = {k}^{ \frac{2}{x} } \times {k}^{ \frac{1}{y} } = {k}^{ \frac{2}{x} + \frac{1}{y} } \\ therefore \: \frac{1}{z} = \frac{1}{y} + \frac{2}{x}$