Question

The value of $7lo{g}_a\frac{16}{15}+5lo{g}_a\frac{25}{24}+3lo{g}_a\frac{81}{80}$ is

• A. $lo{g}_{a3}$
• B. $lo{g}_{a1}$
• C. $lo{g}_{a2}$
• D. $lo{g}_{a5}$

Answer 1

The correct option is C $lo{g}_{a2}$

We have,

$7lo{g}_a\frac{16}{15}+5lo{g}_a\frac{25}{24}+3lo{g}_a\frac{81}{80}$

$\Rightarrow lo{g}_a{\left(\frac{16}{15}\right)}^7+lo{g}_a{\left(\frac{25}{24}\right)}^5+lo{g}_a{\left(\frac{81}{80}\right)}^3$

$\Rightarrow lo{g}_a[{\left(\frac{16}{15}\right)}^7\times {\left(\frac{25}{24}\right)}^5\times {\left(\frac{81}{80}\right)}^3]$

$\Rightarrow lo{g}_a(\frac{{16}^3\times {16}^4}{5^7\times 3^7}\times \frac{5^5\times 5^5}{8^5\times 3^5}\times \frac{3^6\times 3^6}{{16}^3\times 5^3})$

$\Rightarrow lo{g}_a(\frac{{16}^4}{1\times 1}\times \frac{1}{8^5\times 1}\times \frac{1}{1})$

$\Rightarrow lo{g}_a\left(\frac{2^4\times 8^4}{8^5}\right)$

$\Rightarrow lo{g}_a\left(\frac{16}{8}\right)$

$\Rightarrow lo{g}_{a}2$

Hence, C is the answer.