Solve simple logarithmic equations by combining logs and rewriting in exponential form
Problem
Solve \(\ln(x)~+~\ln(x~-~3)~=~\ln(10)\) by combining logarithms and checking every candidate against the original log domain.
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What this problem is really about
Record the original log domain before changing the equation. Condense the sum into one logarithm, use one-to-one behavior to equate its positive argument with the other side's argument, and solve the resulting algebraic equation. Every algebraic root is only a candidate until it passes the original positivity conditions.
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