উদাহরণ ৫৷ প্রমান করো [tex]{\log _5}\sqrt {5\sqrt {5\sqrt {5 \ldots \ldots \infty } } } = 1[/tex] [H.S’93]
প্রমান: মনে করি [tex]x = \sqrt {5\sqrt {5\sqrt {5 \ldots \ldots \infty } } } [/tex]
অতএব
[tex]\begin{array}{l}
x = \sqrt {5\sqrt {5\sqrt {5 \ldots \ldots \infty } } } \\
\Rightarrow {x^2} = 5\sqrt {5\sqrt {5\sqrt {5 \ldots \ldots \infty } } } \\
\Rightarrow {x^2} = 5x\\
\Rightarrow {x^2} - 5x = 0\\
\Rightarrow x\left( {x - 5} \right) = 0\\
\Rightarrow x = 5\left( {x \ne 0} \right)
\end{array}[/tex]
[tex]\begin{array}{l}
{\log _5}\sqrt {5\sqrt {5\sqrt {5 \ldots \ldots \infty } } } \\
= {\log _5}x\\
= {\log _5}5\\
= 1\left( {proved} \right)
\end{array}[/tex]
