If (a/b)^{x − 1} = (b/a)^{x − 3}, find x.

Given data

• (a/b)^{x − 1} = (b/a)^{x − 3}.

Concept / Approach

• Use (b/a) = (a/b)^{−1} to rewrite both sides with the same base.

Step-by-step simplification

Right side: (b/a)^{x − 3} = ( (a/b)^{−1} )^{x − 3} = (a/b)^{−(x − 3)}.Equate exponents (same positive base a/b): x − 1 = −(x − 3).x − 1 = −x + 3 ⇒ 2x = 4 ⇒ x = 2.

Verification

Left: (a/b)^{1}; Right: (b/a)^{−1} = (a/b)^{1} ⇒ equal.

Common pitfalls

• Treating (b/a) as independent base without converting; leads to messy logs.

Final Answer

2.