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

Verified step by step guidance

1. 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: \( \log \left( \frac{3x + 5}{2x + 4} \right) = 0 \).
2. 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 \( \frac{3x + 5}{2x + 4} = 10^0 \).
3. Simplify the right side since \( 10^0 = 1 \), so the equation becomes \( \frac{3x + 5}{2x + 4} = 1 \).
4. Solve the resulting equation by cross-multiplying: \( 3x + 5 = 2x + 4 \). Then isolate \( x \) by subtracting \( 2x \) and 4 from both sides.
5. 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.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, especially the subtraction rule log(a) - log(b) = log(a/b), is essential. This allows combining or simplifying logarithmic expressions to solve equations more easily.

Solving Logarithmic Equations

Solving logarithmic equations involves rewriting the equation using log properties, then converting the logarithmic form to an exponential form to isolate the variable. This step is crucial to find exact solutions.

Domain Restrictions in Logarithms

Since logarithms are only defined for positive arguments, it is important to determine the domain restrictions by setting the inside of each log greater than zero. This ensures that solutions are valid and do not produce undefined expressions.