Solve each equation. Give solutions in exact form. See Examples 5–9. log₂ (2x - 3) + log₂ (x + 1) = 1

Verified step by step guidance

1. Recall the logarithmic property that allows you to combine the sum of two logarithms with the same base into a single logarithm: \(\log_b A + \log_b B = \log_b (A \times B)\). Apply this to the left side of the equation: \(\log_2 (2x - 3) + \log_2 (x + 1) = \log_2 \big((2x - 3)(x + 1)\big)\).
2. Rewrite the equation using the combined logarithm: \(\log_2 \big((2x - 3)(x + 1)\big) = 1\).
3. Recall that \(\log_b M = N\) means \(b^N = M\). Use this to rewrite the equation without logarithms: \$2^1 = (2x - 3)(x + 1)$.
4. Simplify the right side by expanding the product: \((2x - 3)(x + 1) = 2x \cdot x + 2x \cdot 1 - 3 \cdot x - 3 \cdot 1 = 2x^2 + 2x - 3x - 3 = 2x^2 - x - 3\).
5. Set up the quadratic equation: \$2 = 2x^2 - x - 3\(. Move all terms to one side to get \)0 = 2x^2 - x - 3 - 2\(, which simplifies to \)0 = 2x^2 - x - 5$. This quadratic equation can now be solved using the quadratic formula or factoring.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, especially the product rule, is essential. The product rule states that log_b(A) + log_b(B) = log_b(AB), allowing the combination of multiple logarithmic terms into a single log expression for easier solving.

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, which can then 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 the solutions satisfy the domain restrictions, ensuring the expressions inside the logs are greater than zero.