Question

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Answer 1

$Given \: \: \: Question \: \: Is \: \\ \\ 2 {}^{x} = 3 {}^{y} = (12 ){}^{z} \: \: \: \: show \: that \: \: \: \frac{1}{z} = \frac{1}{y} + \frac{2}{x} \\ \\ Answer \: \\ \\ let \: \: \: \: 2 {}^{x} = 3 {}^{y} = (12) {}^{z} \: \: = k \: \: \\ \\ 2 {}^{x} = k \: \: \: \: \: \: \: \: 3 {}^{y} = k \: \: \: \: \: \: \: and \: \: \: \: \: (12 ){}^{z} = k \\ \\ 2 = k {}^{ \frac{1}{x} } \: \: \: ... \: \: Equation \: \: \: i \\ \\ 3 = k {}^{ \frac{1}{y} } \: \: ...Equation \: \: \: ii \\ \\ 12 = k {}^{ \frac{1}{z} } \: \: \: ...Equation \: \: \: iii \: \\ \\ \\ 12 = k {}^{ \frac{1}{z} } \\ \\ 4 \times 3 = k {}^{ \frac{1}{z} } \\ \\ 2 {}^{2} \times 3 = k {}^{ \frac{1}{z} } \\ \\ k {}^{ \frac{2}{x} } \times k {}^{ \frac{1}{y} } = k {}^{ \frac{1}{z} } \\ \\ k {}^{ \frac{1}{z} } = k {}^{( \frac{1}{y} + \frac{2}{x} )} \\ \\ now \: compare \: powers \: \: \: of \: k \: we \: have \\ \\ \frac{1}{z} = \frac{2}{x} + \frac{1}{y} \: \: \: \: hence \: proved \: \: \: \\ \\ \\ Note \: \\ \\ if \: \: \: b {}^{n} = q \: \: \: \: \: \: \: \: \: \: then \: \: \: \: \: \: b = q {}^{ \frac{1}{n} }$