Combine logs: \( \log_2((x + 3)(x - 1)) = 3 \)

["# Solving Combine Logarithmic Equations: How to Solve ( \log_2((x + 3)(x - 1)) = 3 )", "Understanding how to solve logarithmic equations is essential in algebra, particularly when logarithms combine expressions inside their arguments. One common type of problem combines logarithms using the product rule, as seen in the equation:", "[\n\log_2((x + 3)(x - 1)) = 3\n]", "In this article, we’ll walk through step-by-step techniques to solve this type of logarithmic equation, explain key concepts, and highlight best practices for combining and solving logs in algebra.", "---", "## What Does ( \log_2((x + 3)(x - 1)) = 3 ) Mean?", "The equation expresses that the logarithm base 2 of the product ((x + 3)(x - 1)) equals 3. Using the property that ( \log_b(A \cdot B) = \log_b A + \log_b B ), we can rewrite this equation more simply:", "[\n\log_2((x + 3)(x - 1)) = \log_2(x + 3) + \log_2(x - 1) = 3\n]", "This step allows us to exponentiate both sides to eliminate the logarithm.", "---", "## Step 1: Eliminate the Logarithm Using Exponential Form", "Since the base 2 log equals 3, convert to exponential (power) form:", "[\n(x + 3)(x - 1) = 2^3 = 8\n]", "Now the equation becomes a simple quadratic:", "[\n(x + 3)(x - 1) = 8\n]", "---", "## Step 2: Expand and Form a Quadratic Equation", "Multiply the left-hand side:", "[\nx^2 - x + 3x - 3 = x^2 + 2x - 3\n]", "So, the equation is:", "[\nx^2 + 2x - 3 = 8\n]", "Subtract 8 from both sides:", "[\nx^2 + 2x - 11 = 0\n]", "---", "## Step 3: Solve the Quadratic Equation", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, (a = 1), (b = 2), (c = -11). Compute the discriminant:", "[\n\Delta = 2^2 - 4(1)(-11) = 4 + 44 = 48\n]", "So,", "[\nx = \frac{-2 \pm \sqrt{48}}{2} = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3}\n]", "Thus, the two potential solutions are:", "[\nx_1 = -1 + 2\sqrt{3}, \quad x_2 = -1 - 2\sqrt{3}\n]", "---", "## Step 4: Check Domain Restrictions", "Because logarithms are only defined for positive arguments, the expressions inside the log must be greater than zero:", "[\n(x + 3)(x - 1) > 0\n]", "Factor the quadratic inequality:", "[\n(x + 3)(x - 1) > 0\n]", "The roots are (x = -3) and (x = 1). Using a sign chart:", "- On ( (-\infty, -3) ), both factors negative → product positive\n- On ( (-3, 1) ), one factor negative, one positive → product negative\n- On ( (1, \infty) ), both positive → product positive", "So the solution domain is:", "[\nx < -3 \quad \ ext{or} \quad x > 1\n]", "---", "## Step 5: Validate Solutions Against the Domain", "Check each root:", "- ( x = -1 + 2\sqrt{3} \approx -1 + 3.464 = 2.464 > 1 ) → valid\n- ( x = -1 - 2\sqrt{3} \approx -1 - 3.464 = -4.464 < -3 ) → valid", "Both solutions lie in the domain and satisfy the original logarithmic equation.", "---", "## Final Answer", "The solutions to ( \log_2((x + 3)(x - 1)) = 3 ) are:", "[\n\boxed{x = -1 + 2\sqrt{3} \quad \ ext{and} \quad x = -1 - 2\sqrt{3}}\n]", "---", "## Summary", "- Use logarithmic identities to combine logs before exponentiation\n- Always convert to exponential form to remove logarithms\n- Solve resulting algebraic equations (often quadratics)\n- Check solutions against domain conditions to eliminate extraneous roots\n- Understand inequalities to restrict valid values", "Mastering combine logs in equations like this strengthens algebraic problem-solving and prepares you for advanced math topics, including exponential models and real-world applications of logarithmic scaling.", "---", "If you're studying logarithmic equations, practice identifying combined logs, apply exponent rules carefully, and rigorously verify solutions—this approach applies equally to more complex expressions such as ( \log_2((x + 3)(x - 1)(x + 2)) = 4 ) or similar problems."]