In Exercises 74–79, solve each logarithmic equation. log2 (x+3) + log2 (x-3) =4

Verified step by step guidance

1. Use the property of logarithms that states \( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \) to combine the logarithms: \( \log_2((x+3)(x-3)) = 4 \).
2. Recognize that \( (x+3)(x-3) \) is a difference of squares, which simplifies to \( x^2 - 9 \).
3. Rewrite the equation using the property of logarithms: \( \log_2(x^2 - 9) = 4 \).
4. Convert the logarithmic equation to its exponential form: \( x^2 - 9 = 2^4 \).
5. Solve the equation \( x^2 - 9 = 16 \) for \( x \) by adding 9 to both sides and then taking the square root.

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(m) + log_b(n) = log_b(m*n), and the power rule, which allows us to express log_b(m^k) as k*log_b(m). These properties help simplify logarithmic expressions and facilitate solving equations.

Exponential Form

Logarithmic equations can often be solved by converting them into exponential form. For example, if log_b(a) = c, then a = b^c. This transformation is crucial for isolating the variable in logarithmic equations, allowing us to express the equation in a more manageable format.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive real numbers. In the equation log2(x+3) + log2(x-3) = 4, both x+3 and x-3 must be greater than zero. This means x must be greater than 3, which is important to consider when solving the equation to ensure that the solutions are valid within the domain.