Question

If $a^{3-x}{b}^{5x}=a^{x+5}{b}^{3x}$, then the value of $x\log \left(\frac{b}{a}\right)$ is equal to

• A. $\log a$
• B. $\log b$
• C. $\log x$
• D. $1$

Answer 1

The correct option is A $\log a$

$a^{3-x}.b^{5x}=a^{x+5}.b^{3x}$

$\Rightarrow \frac{a^{3-x}}{a^{x+5}}=\frac{b^{3x}}{b^{5x}}$

$\Rightarrow a^{-2(x+1)}=b^{-2x}$
$\Rightarrow a^{-2x}.a^{-2}=b^{-2x}$
$\Rightarrow (\frac{b}{a}{)}^{-2x}=a^{-2}$

$\Rightarrow (\frac{b}{a}{)}^x=a$

$\Rightarrow \log (\frac{b}{a}{)}^x=\log a$

$\Rightarrow x\log \left(\frac{b}{a}\right)=\log a$