উদাহরণ ১৷
প্রমান করো [tex]7\log \frac{{10}}{9} + 3\log \frac{{81}}{{80}} = 2\log \frac{{25}}{{24}} + \log 2[/tex] [H.S '82]
প্রমান:
[tex]\begin{array}{l}
7\log \frac{{10}}{9} + 3\log \frac{{81}}{{80}}\\
= 7\left( {\log 10 - \log 9} \right) + 3\left( {\log 81 - \log 80} \right)\\
= 7\log 10 + 3\log 81 - \left( {7\log 9 + 3\log 80} \right)\\
= 7\log \left( {2 \times 5} \right) + 3\log {3^4} - 7\log {3^2} - 3\log ({2^4} \times 5)\\
= 7\log 2 + 7\log 5 + 12\log 3 - 14\log 3 - 12\log 2 - 3\log 5\\
= - 5\log 2 - 2\log 3 + 4\log 5\\
= 2\log {5^2} + \log 2 - 6\log 2 - 2\log 3\\
= 2\log 25 - 2\log {2^3} - 2\log 3 + \log 2\\
= 2\log \frac{{25}}{{8 \times 3}} + \log 2\\
= 2\log \frac{{25}}{{24}} + \log 2\left( {proved} \right)
\end{array}[/tex]