Solve each equation. Give solutions in exact form. log₂ [(2x + 8)(x + 4)] = 5

Verified step by step guidance

1. Start by recognizing that the equation involves a logarithm with base 2: \(\log_2 \left[(2x + 8)(x + 4)\right] = 5\).
2. Use the property of logarithms that allows you to rewrite the equation in exponential form: if \(\log_b A = C\), then \(A = b^C\). So, rewrite the equation as \((2x + 8)(x + 4) = 2^5\).
3. Calculate the right side exponent: \$2^5\( equals 32, so the equation becomes \)(2x + 8)(x + 4) = 32$.
4. Expand the left side by distributing: multiply \$2x\( by \)x\( and \)4\(, then multiply \)8\( by \)x\( and \)4$, and combine like terms to form a quadratic equation.
5. Set the quadratic equation equal to 32, then move all terms to one side to set the equation to zero. Solve the quadratic equation using factoring, completing the square, or the quadratic formula to find the exact values of \(x\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product rule log_b(MN) = log_b(M) + log_b(N), is essential. These properties allow you to simplify or expand logarithmic expressions, making it easier to solve equations involving logs.

Solving Exponential Equations

After rewriting the logarithmic equation in exponential form, solving the resulting polynomial or algebraic equation is necessary. This involves isolating the variable and using algebraic techniques like factoring or the quadratic formula.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving log equations, it is crucial to check that the solutions make the arguments inside the log positive, discarding any extraneous solutions that do not satisfy this domain restriction.