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1. $If {(\frac{a}{b})}^{x - 1} = {(\frac{b}{a})}^{x - 3},$ then the value of x is

A. $\frac{1}{2}$

B. 1

C. 2

D. $\frac{7}{2}$

Solution:

\begin{aligned} Given {(\frac{a}{b})}^{x - 1} = {(\frac{b}{a})}^{x - 3} \\ \Rightarrow {(\frac{a}{b})}^{x - 1} = {(\frac{a}{b})}^{- (x - 3)} = {(\frac{a}{b})}^{(3 - x)} \\ \Rightarrow x - 1 = 3 - x \\ \Rightarrow 2 x = 4 \\ \Rightarrow x = 2 \end{aligned}

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