If 9(2x−1) = 25 − 5, then the value of x is

Concept

For an exponential equation with the same positive base on both sides (am = an, a > 0, a ≠ 1), the exponents must be equal: m = n. Also recall the power-of-a-power rule (am)n = amn, which lets any base be rewritten as a power of a smaller common base.

Application

1. Simplify the right side: 25 − 5 = 32 − 5 = 27.

2. Rewrite both sides using base 3, since 9 = 32: 9(2x−1) = (32)(2x−1) = 32(2x−1); and 27 = 33.

3. Since the bases now match, equate the exponents: 2(2x−1) = 3.

4. Expand and solve the resulting linear equation: 4x − 2 = 3, so 4x = 5, giving x = 5/4.

Cross-check

Substituting x = 5/4 back: 2x − 1 = 3/2, so 9(3/2) = (9(1/2))3 = 33 = 27, which equals 25 − 5 — confirming x = 5/4 satisfies the original equation.