Question

Solve: ${\log }_{10}\left(\frac{351}{539}\right)+2{\log }_{10}\left(\frac{91}{110}\right)-3{\log }_{10}\left(\frac{39}{110}\right)$

Answer 1

Consider the given expression.

$\Rightarrow {\log }_{10}\left(\frac{351}{539}\right)+2{\log }_{10}\left(\frac{91}{110}\right)-3{\log }_{10}\left(\frac{39}{110}\right)$

$\Rightarrow {\log }_{10}\left(\frac{351}{539}\right)+2{\log }_{10}\left(\frac{91}{110}\right)+3{\log }_{10}\left(\frac{110}{39}\right)$

$\Rightarrow {\log }_{10}\left(\frac{3^3\times 13}{7^2\times 11}\right)+2{\log }_{10}\left(\frac{7\times 13}{2\times 5\times 11}\right)+3{\log }_{10}\left(\frac{2\times 5\times 11}{3\times 13}\right)$

$\Rightarrow {\log }_{10}\left(\frac{3^3\times 13}{7^2\times 11}\right)+{\log }_{10}{\left(\frac{7\times 13}{2\times 5\times 11}\right)}^2+{\log }_{10}{\left(\frac{2\times 5\times 11}{3\times 13}\right)}^3$

$\Rightarrow {\log }_{10}\left(\frac{3^3\times 13}{7^2\times 11}\right)+{\log }_{10}\left(\frac{7^2\times {13}^2}{2^2\times 5^2\times {11}^2}\right)+{\log }_{10}\left(\frac{2^3\times 5^3\times {11}^3}{3^3\times {13}^3}\right)$

$\Rightarrow {\log }_{10}\left(\frac{3^3\times 13\times 7^2\times {13}^2\times 2^3\times 5^3\times {11}^3}{7^2\times 11\times 2^2\times 5^2\times {11}^2\times 3^3\times {13}^3}\right)$

$\Rightarrow {\log }_{10}\left(\frac{3^3\times {13}^3\times 7^2\times 2^3\times 5^3\times {11}^3}{7^2\times {11}^3\times 2^2\times 5^2\times 3^3\times {13}^3}\right)$

$\Rightarrow {\log }_{10}\left(10\right)$

$\Rightarrow 1$

Hence, the value of the above expression is $1$.