উদাহরণ ১০৷ সমাধান করো [tex]{x^{{{\log }_{10}}x}} = 100x[/tex] [H.S’97]
সমাধান:
[tex]\begin{array}{l}
{x^{{{\log }_{10}}x}} = 100x\\
\Rightarrow {\log _{10}}x = {\log _x}(100x)\\
\Rightarrow {\log _{10}}x = {\log _x}100 + {\log _x}x\\
\Rightarrow {\log _{10}}x = {\log _x}{\left( {10} \right)^2} + 1\\
\Rightarrow {\log _{10}}x = 2{\log _x}10 + 1\\
\Rightarrow {\log _{10}}x - 1 = \frac{2}{{{{\log }_{10}}x}}\\
\Rightarrow a - 1 = \frac{2}{a}\left[ {a = {{\log }_{10}}x} \right]\\
\Rightarrow {a^2} - a = 2\\
\Rightarrow {a^2} - a - 2 = 0\\
\Rightarrow {a^2} - 2a + a - 2 = 0\\
\Rightarrow \left( {a - 2} \right)\left( {a + 1} \right) = 0\\
\Rightarrow a = 2,or,a = - 1
\end{array}[/tex]
যখন [tex]a = 2[/tex]
[tex]\begin{array}{l}
{\log _{10}}x = 2\\
\Rightarrow x = {\left( {10} \right)^2} = 100
\end{array}[/tex]
যখন [tex]a = - 1[/tex]
[tex]\begin{array}{l}
{\log _{10}}x = - 1\\
\Rightarrow x = {\left( {10} \right)^{ - 1}} = \frac{1}{{100}}
\end{array}[/tex]
