If \(\mathrm{3^{(x-1)} = 81}\), what is the value of x?

1. INFER the solution strategy

• Given: \(\mathrm{3^{(x-1)} = 81}\)
• Key insight: To solve exponential equations, we need matching bases on both sides
• Strategy: Express 81 as a power of 3, then use the property that equal bases mean equal exponents

2. INFER what power of 3 equals 81

• Calculate powers of 3: \(\mathrm{3^1 = 3}\), \(\mathrm{3^2 = 9}\), \(\mathrm{3^3 = 27}\), \(\mathrm{3^4 = 81}\)
• Therefore: \(\mathrm{81 = 3^4}\)

3. SIMPLIFY by rewriting the equation

• Original: \(\mathrm{3^{(x-1)} = 81}\)
• Rewritten: \(\mathrm{3^{(x-1)} = 3^4}\)

4. INFER that equal bases mean equal exponents

• Since \(\mathrm{3^{(x-1)} = 3^4}\) and the bases are the same
• The exponents must be equal: \(\mathrm{x - 1 = 4}\)

5. SIMPLIFY to solve for x

• \(\mathrm{x - 1 = 4}\)
• \(\mathrm{x = 4 + 1 = 5}\)

Answer: D) 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that both sides need to be expressed with the same base

Students might try to work directly with \(\mathrm{3^{(x-1)} = 81}\) without converting 81 to a power of 3. They might attempt to "take the log" or use other advanced techniques they're not comfortable with, or simply give up when they can't see a clear path forward. This leads to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge: Not remembering the exponent property that equal bases imply equal exponents

Even if students successfully convert 81 to \(\mathrm{3^4}\), they might not know what to do next with \(\mathrm{3^{(x-1)} = 3^4}\). Without knowing this fundamental property, they cannot proceed systematically. This may lead them to select Choice B (3) by incorrectly thinking x = 3 since the base is 3.

The Bottom Line:

This problem tests the fundamental strategy for solving exponential equations: express both sides with matching bases, then equate exponents. Success depends on recognizing this approach and knowing the key exponent property.