If-0-lt-x-lt-1-lt-y-lt-100-lt-z-and-satisfy-these-equations-log-2-xyz-103-1-log-2-x-1-log-2-y-1-log-2-z-1-103-Find-xyz-x-y-z-xy-yz-zx-

Question Number 106288 by ZiYangLee last updated on 04/Aug/20

If 0 < x < 1 < y < 100 < z, and satisfy

these equations :

{\begin{cases} \log_{2} (xyz) = 103 \\ \frac{1}{\log_{2} x} + \frac{1}{\log_{2} y} + \frac{1}{\log_{2} z} = \frac{1}{103} \end{cases}

Find xyz (x + y + z) - xy - yz - zx

Answered by bemath last updated on 04/Aug/20

(1) xyz = 2^{103}

(2) \frac{1}{\log_{2} x} + \frac{1}{\log_{2} (\frac{2^{103}}{xz})} + \frac{1}{\log_{2} (z)} = \frac{1}{103}

\frac{1}{\log_{2} x} + \frac{1}{103 - (\log_{2} x + \log_{2} z)} + \frac{1}{\log_{2} z} = \frac{1}{103}

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