If-2-x-3-y-6-z-find-the-value-of-1-x-1-y-1-z-

Question Number 8998 by tawakalitu last updated on 11/Nov/16

If 2^{x} = 3^{y} = 6^{- z}

find the value of : \frac{1}{x} + \frac{1}{y} + \frac{1}{z}

Answered by Rasheed Soomro last updated on 12/Nov/16

If 2^{x} = 3^{y} = 6^{- z}

find the value of : \frac{1}{x} + \frac{1}{y} + \frac{1}{z}

- - - - - - - - - - - - - - - - - -

2^{x} = 3^{y} = 6^{- z}

\Rightarrow xlog 2 = ylog 3 = - zlog 6

x = - z (\frac{\log 6}{\log 2}), y = - z (\frac{\log 6}{\log 3})

x^{- 1} = {- z (\frac{\log 6}{\log 2})}^{- 1} \Rightarrow \frac{1}{x} = - \frac{\log 2}{zlog 6}

y^{- 1} = {- z (\frac{\log 6}{\log 3})}^{- 1} \Rightarrow \frac{1}{y} = - \frac{\log 3}{zlog 6}

\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = - \frac{\log 2}{zlog 6} - \frac{\log 3}{zlog 6} + \frac{1}{z} = \frac{- \log 2 - \log 3 + \log 6}{zlog 6}

= \frac{{\log 2}^{- 1} + {\log 3}^{- 1} + \log 6}{zlog 6} = \frac{\log (\frac{1}{2} \times \frac{1}{3} \times 6)}{zlog 6} = \frac{\log 1}{zlog 6}

= \frac{0}{zlog 6} = 0

\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0

Commented by tawakalitu last updated on 11/Nov/16

Thank you sir .

Commented by Theara last updated on 19/Nov/16

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