Question

If $a^4\cdot b^5=1$, then the value of ${\log }_a\left(a^5{b}^4\right)$ equals

• A. $\frac{9}{5}$
• B. $4$
• C. $5$
• D. $\frac{8}{5}$

Answer 1

The correct option is A $\frac{9}{5}$

Given that, $a^4\cdot b^5=1$
$\Rightarrow \log a^4\cdot b^5=0$
$\Rightarrow 4{\log }_{a}a+5{\log }_{a}b=0$
$\Rightarrow {\log }_{a}b=\frac{-4{\log }_{a}a}{5}$ ...(1)
$y={\log }_a\left(a^5{b}^4\right)$
$\Rightarrow y=5{\log }_{a}a+4{\log }_{a}b$
$\Rightarrow y=5{\log }_{a}a+4\left(\frac{-4{\log }_{a}a}{5}\right)$ ...{ from 1}
$\Rightarrow y=\frac{9}{5}{\log }_{a}a=\frac{9}{5}$
Ans: A