Question

If $a^{1-2x}.b^{1+2x}=a^{4+x}.b^{4-x};a,b>0\&a\ne b$, then the value of $x$ is

• A. ${\log }_{\left(\frac{b}{a}\right)}\left(ab\right)$
• B. ${\log }_{\left(\frac{a}{b}\right)}\left(ab\right)$
• C. $\frac{1}{2}{\log }_{\left(\frac{a}{b}\right)}\left(ab\right)$
• D. $\frac{1}{2}{\log }_{\left(\frac{b}{a}\right)}\left(ab\right)$

Answer 1

The correct option is A ${\log }_{\left(\frac{b}{a}\right)}\left(ab\right)$

$a^{3(x+1)}=b^{3(x-1)}\Rightarrow 3(x+1)\log a=3(x-1)\log b\Rightarrow x(\log b-\log a)=\log b+\log a\Rightarrow x=\frac{\log b+\log a}{\log b-\log a}\Rightarrow x={\log }_{\left(\frac{b}{a}\right)}\left(ab\right)$