Question

If ${\left(\frac{a}{b}\right)}^{x-1}={\left(\frac{b}{a}\right)}^{x-3}$ find the value of $x$

• A. $2$
• B. $3$
• C. $4$
• D. $1$

Answer 1

The correct option is A $2$

$Now{\left(\frac{a}{b}\right)}^{x-1}={\left(\frac{b}{a}\right)}^{x-3}\to \frac{a^{x-1}}{b^{x-1}}=\frac{b^{x-3}}{a^{x-3}}\to a^{x-1}\times a^{x-3}=b^{x-3}\times b^{x-1}\to a^{x-1+x-3}=b^{x-3+x-1}\to a^{2x-4}=b^{2x-4}\to \therefore a=b\to a^{2x-4}=a^{2x-4}\to \frac{a^{2x-4}}{a^{2x-4}}=1\to a^{2x-4-2x+4}=1\to a^0=1\therefore 2x-4=0\to 2x=4\to x=2$