Evaluate: log(a^2 / (b c)) + log(b^2 / (a c)) + log(c^2 / (a b)). Simplify to a single constant value.
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A0
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B1
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Cabc
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Dab2c2
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ENone of these
Answer
Correct Answer: 0
Explanation
Introduction / Context:We simplify a sum of logarithms involving symmetric rational expressions in a, b, and c. The task is purely algebraic, requiring log combination rules.
Given Data / Assumptions:
- Expression: log(a^2/(bc)) + log(b^2/(ac)) + log(c^2/(ab))
- a, b, c > 0 (so logs are defined).
Concept / Approach:Use log A + log B + log C = log(ABC). Multiply the three rational expressions and simplify the resulting fraction by collecting powers of a, b, c in numerator and denominator.
Step-by-Step Solution:
Product inside one log: (a^2/(bc))·(b^2/(ac))·(c^2/(ab))Numerator: a^2 b^2 c^2Denominator: (b c)(a c)(a b) = a^2 b^2 c^2Therefore the product equals 1Hence the sum equals log(1) = 0Verification / Alternative check:By exponent rules: exponents of a, b, and c in the product are 2 − 1 − 1 = 0 each, confirming that every variable cancels and the product is 1.
Why Other Options Are Wrong:1 would equal log(10) in base 10, not log(1); abc or ab2c2 are not constants and do not reflect the cancellation evident above.
Common Pitfalls:Forgetting to add exponents in the denominator or mishandling cancellations can obscure that the product equals 1 and the log is 0.
Final Answer:0