Problem

Audit the real-log rewrite \(\log(xy)~=~\log(x)~+~\log(y)\). State where it is valid, whether it preserves the full domain \(xy~>~0\), and the full-domain expansion when needed.

Big Picture

What this problem is really about

Audit equality and domain equivalence as separate questions. A positive product can come from two positive factors or two negative factors, while separate real logarithms without absolute values cover only the positive-positive branch. Testing the missing sign case reveals whether absolute values are needed to preserve the original full domain.

Answer choices

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Candidate analysis: log(xy) = log(x) + log(y) is valid when x > 0 and y > 0, but not domain-equivalent on the full domain xy > 0. For that full domain, use log|xy|.

Candidate analysis: log(xy) = log(x) + log(y) is valid when x > 0 and y > 0, and domain-equivalent on the full domain xy > 0. For that full domain, use log(x) + log(y).

Candidate analysis: log(xy) = log(x) + log(y) is valid when x > 0 and y > 0, but not domain-equivalent on the full domain xy > 0. For that full domain, use log|x| + log|y|.

Candidate analysis: log(xy) = log(x) + log(y) is invalid when x > 0 and y > 0, but not domain-equivalent on the full domain xy > 0. For that full domain, use log|x| + log|y|.

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Audit the real-log rewrite \(\log(xy)~=~\log(x)~+~\log(y)\). State where it is valid, whether it preserves the full domain \(xy~>~0\), and the full-domain expansion when needed.

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