Evaluate the difference: log₄₉(16807) − log₉(27). Convert to prime powers first to obtain an exact rational result.

Introduction / Context:We compute two exact logarithms by expressing both bases and arguments as powers of a common prime and then subtract.

Given Data / Assumptions:

• 49 = 7^2, 16807 = 7^5
• 9 = 3^2, 27 = 3^3

Concept / Approach:Use log_{p^m}(p^n) = n/m. Then subtract the two rational numbers obtained.

Step-by-Step Solution:

Verification / Alternative check:Change of base to natural logs gives ln(7^5)/ln(7^2) − ln(3^3)/ln(3^2) = (5/2) − (3/2) as above.

Why Other Options Are Wrong:0 or −1 would require equal or reversed fractions; 3/2 is one of the individual logs, not the difference.

Common Pitfalls:Mistaking 16807 as 7^6 or misreading 49/9 bases leads to wrong fractions. Confirm prime-power factorizations first.

Final Answer:1