Solve the exponential equation for x: 9^(x - 1) = 27^(2x - 4).
Answer and explanation
Correct answer: C. x = 5/2
- Step 1: Express both bases as powers of 3: 9 = 3^2 and 27 = 3^3.
- Step 2: Rewrite: (3^2)^(x - 1) = (3^3)^(2x - 4).
- Step 3: Apply exponent laws: 3^(2x - 2) = 3^(6x - 12).
- Step 4: Equate exponents: 2x - 2 = 6x - 12.
- Step 5: Rearrange: 10 = 4x, so x = 10/4 = 5/2.
Answer
x = 5/2.
Why not the others: 'x = 2' equates bases without applying powers of 3; 'x = 3' makes an arithmetic sign mistake (6x - 2x = 12 + 2); 'x = 7/4' forgets to distribute 3 over -4.
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