Question

If 2^a = 3^b = 6^c then show that c = ab/a + c

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Answer 1

$Answer \: \: \\ \\ 2 {}^{a} = 3 {}^{b} = 6 { }^{c} = k \: \: \: let \\ \\ 2 {}^{a} = k \: \: \: \: 3 {}^{b} = k \: \: \: \: \: 6 {}^{c} = k \\ \\ 2 = k {}^{ \frac{1}{a} } ....Equation \: \: \: \:i \: \: \\ \\ 3 = k {}^{ \frac{1}{b} } ....Equation \: \: \: ii \\ \\ 6 = k {}^{ \frac{1}{c} } ... \: Equation \: \: \: iii \\ \\ 6 = k {}^{ \frac{1}{c} } \\ \\ 3 \times 2 = k {}^{ \frac{1}{c} } \\ \\ k {}^{ \frac{1}{b} } \times k {}^{ \frac{1}{a} } = k {}^{ \frac{1}{c} } \\ \\ compare \: \: powers \: of \: k \: we \: have \\ \\ \frac{1}{b} + \frac{1}{a} = \frac{1}{c} \\ \\ \frac{a + b}{ab} = \frac{1}{c} \\ \\ c = \frac{ab}{a + b} \: \: \: hence \: \: proved$