Error message

  • Notice: Undefined index: und in eval() (line 15 of /home/bstudent/public_html/bsthirteen/modules/php/php.module(80) : eval()'d code).
  • Notice: unserialize() [function.unserialize]: Error at offset 0 of 9 bytes in profile_user_load() (line 257 of /home/bstudent/public_html/bsthirteen/modules/profile/profile.module).

Problem 0015 | Logarithm

উদাহরণ ১৫৷\frac{{\log x}}{{ry - qz}} = \frac{{\log y}}{{pz - rx}} = \frac{{\log z}}{{qx - py}} হলে দেখাও যে {x^p}{y^q}{z^r} = 1 [H.S ‘92]

সমাধান:

মনে করি \frac{{\log x}}{{ry - qz}} = \frac{{\log y}}{{pz - rx}} = \frac{{\log z}}{{qx - py}} = k

অতএব

\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}" alt="\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}"/>

অনুরূপে

\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}" alt="\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}"/>

করে পাই

\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}" alt="\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}"/>