Solve each equation. Give solutions in exact form. See Examples 5–9. log_3 [(x + 5)(x - 3)] = 2

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The equation log_b(a) = c means that b raised to the power of c equals a (b^c = a). Understanding this relationship is crucial for solving logarithmic equations, as it allows us to rewrite the logarithmic expression in exponential form.

Properties of Logarithms

Logarithms have several key properties that simplify their manipulation. For instance, the product property states that log_b(mn) = log_b(m) + log_b(n), and the power property states that log_b(m^k) = k * log_b(m). These properties are essential for breaking down complex logarithmic expressions into simpler components.

Solving Quadratic Equations

Many logarithmic equations can lead to quadratic equations when rewritten in exponential form. A quadratic equation is typically in the form ax^2 + bx + c = 0. Solving these equations often involves factoring, using the quadratic formula, or completing the square, which are fundamental techniques in algebra.