In Exercises 74–79, solve each logarithmic equation. log4 (2x+1) = log4 (x-3) + log4 (x+5)

Verified step by step guidance

1. Recognize that the equation involves logarithms with the same base, which allows us to use the properties of logarithms to simplify.
2. Use the property of logarithms that states \( \log_b A + \log_b B = \log_b (A \cdot B) \) to combine the right side: \( \log_4 (x-3) + \log_4 (x+5) = \log_4 ((x-3)(x+5)) \).
3. Set the expressions inside the logarithms equal to each other since the logarithms themselves are equal: \( 2x + 1 = (x-3)(x+5) \).
4. Expand the right side of the equation: \( (x-3)(x+5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15 \).
5. Rearrange the equation to form a quadratic equation: \( 2x + 1 = x^2 + 2x - 15 \) becomes \( 0 = x^2 + 2x - 15 - 2x - 1 \).

Key Concepts

Logarithmic Properties

Understanding the properties of logarithms is essential for solving logarithmic equations. Key properties include the product rule, which states that log_b(mn) = log_b(m) + log_b(n), and the quotient rule, which states that log_b(m/n) = log_b(m) - log_b(n). These properties allow us to combine or separate logarithmic expressions, facilitating the solution process.

Exponential Form

Logarithmic equations can often be solved by converting them into exponential form. For example, if log_b(a) = c, it can be rewritten as b^c = a. This transformation is crucial for isolating variables and simplifying the equation, making it easier to find the solution.

Domain of Logarithmic Functions

When solving logarithmic equations, it is important to consider the domain restrictions. The argument of a logarithm must be positive; thus, any solutions must satisfy the conditions that 2x + 1 > 0, x - 3 > 0, and x + 5 > 0. This ensures that the solutions are valid within the context of the logarithmic function.