Interpret and evaluate: log₃₂(8) + log₂₄₃(3^7) − log₃₆(1296). Combine exact values to obtain an integer result.

Introduction / Context:Each logarithm can be rewritten using prime powers to obtain exact rational values. The goal is to compute each term exactly and then sum.

Given Data / Assumptions:

• Interpretation: log base 32 of 8, plus log base 243 of 3^7, minus log base 36 of 1296.
• All logs are well-defined.

Concept / Approach:Use a = p^m, b = p^n ⇒ log_a b = n/m. Recognize 32 = 2^5, 8 = 2^3; 243 = 3^5, 3^7 is clear; 1296 = 36^2.

Step-by-Step Solution:

Verification / Alternative check:Convert each to natural logs: ln 8/ln 32 + ln 3^7/ln 243 − ln 1296/ln 36; simplifying exponents yields the same rational values, confirming the result.

Why Other Options Are Wrong:3, 2, and 1 would require different exponents or bases; the exact arithmetic shows cancellation to zero.

Common Pitfalls:Misreading bases or exponents (e.g., treating 243 as 3^4) changes ratios and produces incorrect sums. Carefully factor numbers into prime powers.

Final Answer:0