Number System — If 3^n = 81, determine the value of n. Show the reasoning using powers of 3 and exact equality (no logarithms needed).

Introduction / Context:
This is a direct exponent matching problem. The number 81 is a well-known power of 3, so recognizing the relevant exponent avoids longer calculations and ensures an exact answer quickly.

Given Data / Assumptions:

• 3^n = 81.
• n is a real number; we look for the integer that satisfies the equality.
• Basic exponent rules apply.

Concept / Approach:
Rewrite 81 as a power of 3. Using the small powers of 3 helps: 3^2 = 9, 3^3 = 27, 3^4 = 81. Matching the base and comparing exponents yields the solution immediately without logarithms.

Step-by-Step Solution:
1) Recall powers of 3: 3^2 = 9, 3^3 = 27, 3^4 = 81.2) Since 81 = 3^4, set 3^n = 3^4.3) With equal positive bases, exponents are equal: n = 4.

Verification / Alternative check:
Compute 3^4 explicitly: 3*3*3*3 = 9*9 = 81. The equality holds exactly, confirming n = 4.

Why Other Options Are Wrong:
2 and 3 are too small (3^2 = 9, 3^3 = 27); 6 and 8 are too large (3^6 = 729, 3^8 = 6561). None of these equal 81.

Common Pitfalls:
Confusing 81 with 3^3; mixing up 9 and 81; attempting unnecessary logarithms that increase the chance of arithmetic slips.

Final Answer:
4