উদাহরণ ১৫৷[tex]\frac{{\log x}}{{ry - qz}} = \frac{{\log y}}{{pz - rx}} = \frac{{\log z}}{{qx - py}}[/tex] হলে দেখাও যে [tex]{x^p}{y^q}{z^r} = 1[/tex] [H.S ‘92]
সমাধান:
মনে করি [tex]\frac{{\log x}}{{ry - qz}} = \frac{{\log y}}{{pz - rx}} = \frac{{\log z}}{{qx - py}} = k[/tex]
অতএব
[tex]\begin{array}{l}
\log x = k\left( {ry - qz} \right)\\
\Rightarrow p\log x = kp\left( {ry - qz} \right)\\
\Rightarrow \log {x^p} = k\left( {pry - pqz} \right) \to \left( 1 \right)
\end{array}[/tex]
অনুরূপে
[tex]\begin{array}{l}
\log {y^q} = kq\left( {pz - rx} \right) = k\left( {pqz - qrx} \right) \to \left( 2 \right)\\
\log {z^r} = kr\left( {qx - py} \right) = k\left( {qrx - pry} \right) \to \left( 3 \right)
\end{array}[/tex]
[tex]\left( 1 \right) + \left( 2 \right) + \left( 3 \right)[/tex] করে পাই
[tex]\begin{array}{l}
\log {x^p} + \log {y^q} + \log {z^r} = k\left( {rpy - qpz + pqz - qrx + qrx - rpy} \right)\\
\Rightarrow \log \left( {{x^p}{y^q}{z^r}} \right) = 0\\
\Rightarrow {x^p}{y^q}{z^r} = 1\left( {proved} \right)
\end{array}[/tex]
