Question

Simplify $7log\left(\frac{32}{15}\right)+5log\left(\frac{25}{24}\right)+3log\left(\frac{81}{160}\right)$

Answer 1

Given expression:

$7log\left(\frac{32}{15}\right)+5log\left(\frac{25}{24}\right)+3log\left(\frac{81}{160}\right)$

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

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

$=7(5log2-log3-log5)+5(2log5-log3-3log2)+3(4log3-log5-5log2)$

$=35log2-7log3-7log5+10log5-5log3-15log2+12log3-3log5-15log2$

$=35log2-15log2-15log2-7log\left(3\right)-5log\left(3\right)+12log\left(3\right)-7log\left(5\right)+10log\left(5\right)-3log\left(5\right)$

$=5log2$