Question

If $a^4.b^5=1$ 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,
$a^4.b^5=1$ ...................(1)
${\log }_a(a^5.b^4)$
$=\frac{\log (a^5.b^4)}{\log a}$
$=\frac{\log a^5+\log b^4}{\log a}$
$=\frac{5\log a+4\log b}{\log a}$
$=5+4\frac{\log b}{\log a}$...............(2)
Using equation (1),
$a^4.b^5=1$
${\log }_e(a^4.b^5)=0$
$4{\log }_a+5\log b=0$
$\Rightarrow \frac{\log b}{\log a}=-\frac{4}{5}$
Putting this in equation (2),
$5+4(-\frac{4}{5})$
$\frac{9}{5}$