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

Verified step by step guidance

1. Recall the logarithmic property that states \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). Apply this to the left side of the equation to combine the logarithms: \( \ln(5 + 4x) - \ln(3 + x) = \ln \left( \frac{5 + 4x}{3 + x} \right) \).
2. Rewrite the equation using the combined logarithm: \( \ln \left( \frac{5 + 4x}{3 + x} \right) = \ln 3 \).
3. Since the natural logarithm function \( \ln(x) \) is one-to-one, set the arguments equal to each other: \( \frac{5 + 4x}{3 + x} = 3 \).
4. Solve the resulting rational equation for \( x \) by cross-multiplying: \( 5 + 4x = 3(3 + x) \). Then expand and simplify the equation to isolate \( x \).
5. Check your solution(s) by substituting back into the original logarithmic expressions to ensure the arguments of all logarithms are positive, since the domain of \( \ln(x) \) requires \( x > 0 \).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, especially the subtraction rule ln(a) - ln(b) = ln(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 in exponential form after isolating the logarithm. This step converts the problem into an algebraic equation that can be solved for the variable.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving equations involving logs, it is crucial to check that solutions do not make any log argument zero or negative, ensuring valid solutions.