Expand a logarithm of a product without losing its domain
Problem
Assume \(b~>~0\), \(b~\ne~1\), and \(x~>~0\). Expand \(\log_b(6x)\) using the product law and state the validity domain.
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What this problem is really about
Separate only genuine multiplicative factors inside the logarithm; the product law turns those factors into a sum of logs. Before rewriting, solve where the original argument is positive, then confirm that every separated argument is positive on exactly that domain. A correct identity includes both its algebra and its validity conditions.
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