Solve each equation. Give solutions in exact form. log₃ [(x + 5)(x - 3)] = 2

Verified step by step guidance

1. Recognize that the equation is a logarithmic equation with base 3: \(\log_3 \left[(x + 5)(x - 3)\right] = 2\).
2. Use the definition of logarithm to rewrite the equation in exponential form: \((x + 5)(x - 3) = 3^2\).
3. Simplify the right side: \$3^2 = 9\(, so the equation becomes \)(x + 5)(x - 3) = 9$.
4. Expand the left side using the distributive property: \(x^2 - 3x + 5x - 15 = 9\), which simplifies to \(x^2 + 2x - 15 = 9\).
5. Bring all terms to one side to set the quadratic equation to zero: \(x^2 + 2x - 15 - 9 = 0\), which simplifies to \(x^2 + 2x - 24 = 0\). Then solve this quadratic equation for \(x\).

Key Concepts

Properties of Logarithms

Logarithmic properties, such as the product rule, allow combining or expanding logarithmic expressions. For example, log_b(MN) = log_b(M) + log_b(N). Understanding these properties helps simplify or rewrite equations involving logarithms to isolate variables.

Definition of Logarithms and Exponentials

A logarithm log_b(A) = C means that b raised to the power C equals A (b^C = A). This definition is essential for converting logarithmic equations into exponential form, which often makes solving for the variable more straightforward.

Solving Quadratic Equations

After rewriting the logarithmic equation in exponential form, the resulting equation may be quadratic. Knowing how to solve quadratic equations using factoring, completing the square, or the quadratic formula is necessary to find exact solutions.