Question

If $a^4\cdot b^5=1$, then the value of ${\log }_a{a}^5{b}^4$ equals
(where $a,b\in R^{+}$ and $a\ne 1$)

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

Answer 1

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

$a^4\cdot b^5=1\Rightarrow {\log }_a{a}^4{b}^5=0\Rightarrow 4+5{\log }_{a}b=0\Rightarrow {\log }_{a}b=-\frac{4}{5}\dots \left(1\right)$

Now, ${\log }_a{a}^5{b}^4=5+4{\log }_{a}b$
From $\left(1\right)$,
${\log }_a{a}^5{b}^4=5-\frac{16}{5}=\frac{9}{5}$