Expand a logarithm of a quotient and preserve its maximal domain
Problem
Expand \(\log_b(x/7)\) using the quotient law and state its maximal real domain.
Big Picture
What this problem is really about
For a logarithm of a quotient, first require the entire quotient to be positive and its denominator to be nonzero. On that domain, the quotient law becomes the numerator's logarithm minus the denominator's logarithm. Comparing the original and separated domains prevents an algebraically familiar rewrite from admitting or losing inputs.
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