Question

সরল কর: $7\log_{10}\frac{10}{9}-2\log_{10}\frac{25}{24}+3\log_{10}\frac{81}{80}$

Answer 1

$7\log_{10}\frac{10}{9}-2\log_{10}\frac{25}{24}+3\log_{10}\frac{81}{80}$

$=\log_{10}\left(\frac{10}{9}\right)^7-\log_{10}\left(\frac{25}{24}\right)^2+\log_{10}\left(\frac{81}{80}\right)^3$

$=\log_{10}\left\lbrace\left(\frac{10}{9}\right)^7\div\left(\frac{25}{24}\right)^2\times\left(\frac{81}{80}\right)^3\right\rbrace$

$=\log_{10}\left\lbrace\left(\frac{2\times5}{3^2}\right)^7\div\left(\frac{5^2}{2^3\times3}\right)^2\times\left(\frac{3^4}{2^4\times5}\right)^3\right\rbrace$

$=\log_{10}\left(\frac{2^7\times5^7}{3^{14}}\div\frac{5^4}{2^6\times3^2}\times\frac{3^{12}}{2^{12}\times5^3}\right)$

$=\log_{10}\left(\frac{2^7\times5^7}{3^{14}}\times\frac{2^6\times3^2}{5^4}\times\frac{3^{12}}{2^{12}\times5^3}\right)$

$=\log_{10}\left(\frac{2^{13}\times3^{14}\times5^7}{2^{12}\times3^{14}\times5^7}\right)$

$=\log_{10}2$ [Answer]