Solve each equation. Give solutions in exact form. See Examples 5–9. . log₅ (x + 2) + log₅ (x - 2) = 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: \(\log_b A + \log_b B = \log_b (A \times B)\). Apply this to combine the left side: \(\log_5 (x + 2) + \log_5 (x - 2) = \log_5 \big((x + 2)(x - 2)\big)\).
2. Rewrite the equation using the combined logarithm: \(\log_5 \big((x + 2)(x - 2)\big) = 1\).
3. Use the definition of logarithm to rewrite the equation in exponential form: if \(\log_b M = N\), then \(M = b^N\). So, \( (x + 2)(x - 2) = 5^1\).
4. Simplify the left side by expanding the product: \((x + 2)(x - 2) = x^2 - 4\). Then set the equation: \(x^2 - 4 = 5\).
5. Solve the quadratic equation \(x^2 - 4 = 5\) by isolating \(x^2\) and then taking the square root of both sides, remembering to consider both positive and negative roots. Also, check the solutions to ensure they do not make the arguments of the original logarithms negative or zero.

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 logarithmic terms with the same base into a single logarithm.

Solving Logarithmic Equations

Solving logarithmic equations involves rewriting the equation in exponential form after isolating the logarithm. This step helps convert 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 like log_b(x + 2), the expressions inside the logs must be greater than zero, which restricts the possible solutions and must be checked after solving.