Simplify the logarithmic expression: log(75/16) − 2·log(5/9) + log(32/343). Express the result as a single logarithm and identify its exact simplified form.

Introduction / Context:This problem tests manipulation of logarithms using the product, quotient, and power rules. We combine all terms into a single logarithm and then simplify the numeric argument exactly.

Given Data / Assumptions:

• Expression: log(75/16) − 2·log(5/9) + log(32/343)
• All logs share the same base.

Concept / Approach:Use log A − 2·log B + log C = log( A·C / B^2 ). Evaluate the fraction precisely and reduce to lowest terms.

Step-by-Step Solution:

Verification / Alternative check:Prime-factor view: 486 = 2·3^5 and 343 = 7^3, so the exact simplest form is log( (2·3^5) / 7^3 ). No further cancellation exists, confirming the simplified form.

Why Other Options Are Wrong:Values like log 2, 2 log 2, log 3, or log 5 would require the fraction to simplify to 2, 4, 3, or 5 respectively, which it does not; 486/343 ≈ 1.417 is distinct.

Common Pitfalls:Missing the square on the middle term (because of the coefficient 2) or dropping factors during fraction reduction can lead to incorrect constants like log 2 or log 3. Carefully track numerators and denominators.

Final Answer:log(486/343)