Problem

Audit the proposed real-log rewrite \(\log(x~+~y)~=~\log(x)~+~\log(y)\). State whether such a law exists and test it with \(x~=~y~=~1\).

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Candidate analysis: log(x + y) = log(x) + log(y) is valid by the log-of-a-sum law. For example, x = y = 1 gives log(2), not 0.

Candidate analysis: log(x + y) = log(x) + log(y) is valid when x > 0 and y > 0: there is no log-of-a-sum law. For example, x = y = 1 gives log(2), not 0.

Candidate analysis: log(x + y) = log(x) + log(y) is invalid: there is no log-of-a-sum law. For example, x = y = 1 gives log(2), not 0.

Candidate analysis: log(x + y) = log(x) + log(y) is invalid: there is no log-of-a-sum law. For example, x = y = 1 gives 0 on both sides.

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Audit the proposed real-log rewrite \(\log(x~+~y)~=~\log(x)~+~\log(y)\). State whether such a law exists and test it with \(x~=~y~=~1\).

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