Question

Simplify $7\log \frac{16}{15}+5\log \frac{25}{24}+3\log \frac{81}{80}$ upto $2$ decimal places

Answer 1

Given expression
$=7\log \left(\frac{2^4}{3\times 5}\right)+5\log \left(\frac{5^2}{3\times 2^3}\right)+3\log \left(\frac{3^4}{5\times 2^4}\right)$

$=7[\log 2^4-\log (3\times 5\left)\right]+5[\log 5^2-\log (3\times 2^3\left)\right]+3[\log 3^4-\log (5\times 2^4\left)\right]$

$[\because \log \frac{a}{b}=\log a-\log b,\log a^m=m\log a,{\log }_{a}a=1]$
$=7[4\log 2-\log 3-\log 5]+5[2\log 5-\log 3-3\log 2]+3[4\log 3-\log 5-4\log 2]$

$=\log 2=0.3010$

$=0.30$ (upto 2 decimal places)
Ans: 0.30