উদাহরণ ৩৷
যদি [tex]\frac{{\log x}}{{y - z}} = \frac{{\log y}}{{z - x}} = \frac{{\log z}}{{x - y}}[/tex] হয় তবে দেখাও যে [tex]{x^x}{y^y}{z^z} = 1[/tex] [H.S'2000]
প্রমান:
ধরি
[tex]\begin{array}{l}
\frac{{\log x}}{{y - z}} = \frac{{\log y}}{{z - x}} = \frac{{\log z}}{{x - y}} = k\\
\log x = k\left( {y - z} \right),\log y = k\left( {z - x} \right),\log z = k\left( {x - y} \right)
\end{array}[/tex]
[tex]\begin{array}{l}
x\log x = xk\left( {y - z} \right)\\
\Rightarrow \log {x^x} = k\left( {xy - xz} \right) \to \left( 1 \right)\\
y\log y = yk(z - x)\\
\Rightarrow \log {y^y} = k(zy - xy) \to (2)\\
z\log z = zk(x - y)\\
\Rightarrow \log {z^z} = k(xz - yz) \to \left( 3 \right)\\
\left( 1 \right) + \left( 2 \right) + \left( 3 \right)\\
\log {x^x} + \log {y^y} + \log {z^z} = k\left( {xy - xz + yz - yx + xz - yz} \right)\\
\Rightarrow \log {x^x}{y^y}{z^z} = k \times 0\\
\Rightarrow \log {x^x}{y^y}{z^z} = 0\\
\Rightarrow {x^x}{y^y}{z^z} = 1\left( {proved} \right)
\end{array}[/tex]
