Question

Solve each equation. Give solutions in exact form. See Examples 5–9. log(3x + 5) - log(2x + 4) = 0

Accepted Answer

Recall the logarithmic property that states $$ \log_a M - \log_a N = \log_a \left( \frac{M}{N} \right) . $$Apply this to the equation $$ \log(3x + 5) - \log(2x + 4) = 0  to $$combine the logs into a single logarithm: $$ \log \left( \frac{3x + 5}{2x + 4} \right) = 0 . $$Use the definition of logarithm: if $$ \log_b A = C $$, then $$ A = b^C . $$Since the base is 10 (common logarithm), rewrite the equation as $$ \frac{3x + 5}{2x + 4} = 10^0 . $$Simplify the right side since $$ 10^0 = 1 $$, so the equation becomes $$ \frac{3x + 5}{2x + 4} = 1 . $$Solve the resulting equation by cross-multiplying: $$ 3x + 5 = 2x + 4 . $$Then isolate $$ x  by $$subtracting $$ 2x $$ and 4 from both sides. Check the solution(s) by substituting back into the original logarithmic expressions to ensure the arguments $$ 3x + 5 $$ and $$ 2x + 4 $$ are positive, since the logarithm is only defined for positive arguments.

Solve each equation. Give solutions in exact form. See Examples 5–9. log(3x + 5) - log(2x + 4) = 0

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Key Concepts

Properties of Logarithms

Solving Logarithmic Equations

Domain Restrictions in Logarithms