Question

Show that $\log \left(\frac{108}{605}\right)=2\log 2+3\log 3-\log 5-2\log 11$

Answer 1

L.H.S$=\log \left(\frac{108}{605}\right)$

$=\log \left(\frac{2\times 2\times 3\times 3\times 3}{5\times 11\times 11}\right)$

$=\log \left(\frac{2^2\times 3^3}{5{11}^2}\right)$

We know that $\log \left(\frac{a}{b}\right)=\log a-\log b$

$=\log 2^2+\log 3^3-\log 5-\log {11}^2$

We know that $\log a^m=m\log a$

$=2\log 2+3\log 3-\log 5-2\log 11$

$=$R.H.S

$\therefore \log \left(\frac{108}{605}\right)=2\log 2+3\log 3-\log 5-2\log 11$

Hence, proved.