\log_2[(x+3)(x-1)] = 3

["Understanding the Equation: log₂[(x + 3)(x – 1)] = 3", "When solving logarithmic equations, clarity and precision are key. One such equation that frequently arises in algebra and advanced precalculus is:", "[\n\log_2[(x + 3)(x - 1)] = 3\n]", "This article breaks down how to solve this equation step by step, explains its mathematical significance, and offers practical tips for learners and educators alike.", "---", "### What Does the Equation Mean?", "The expression\n[\n\log_2[(x + 3)(x - 1)] = 3\n]\ntells us the logarithm (base 2) of the product $(x + 3)(x - 1)$ equals 3. To solve for $x$, we need to eliminate the logarithm and convert the equation into exponential form.", "---", "### Step-by-Step Solution", "1. Convert to Exponential Form\n Recall that $\log_b(A) = C$ implies $A = b^C$. Applying this property:\n [\n (x + 3)(x - 1) = 2^3 = 8\n ]", "2. Expand the Left-Hand Side\n Multiply the binomials:\n [\n (x + 3)(x - 1) = x^2 - x + 3x - 3 = x^2 + 2x - 3\n ]", "So the equation becomes:\n [\n x^2 + 2x - 3 = 8\n ]", "3. Form a Quadratic Equation\n Subtract 8 from both sides:\n [\n x^2 + 2x - 11 = 0\n ]", "4. Apply the Quadratic Formula\n Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 2$, $c = -11$:\n [\n x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-11)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 44}}{2} = \frac{-2 \pm \sqrt{48}}{2}\n ]", "Simplify $\sqrt{48} = 4\sqrt{3}$:\n [\n x = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3}\n ]", "5. Check Validity of Solutions\n Logarithmic functions are only defined for positive arguments, so we require:\n [\n (x + 3)(x - 1) > 0\n ]", "The product $(x + 3)(x - 1)$ is positive when $x < -3$ or $x > 1$.\n Approximate the solutions:\n - $x = -1 + 2\sqrt{3} \approx -1 + 3.464 = 2.464$ (>1 → valid)\n - $x = -1 - 2\sqrt{3} \approx -1 - 3.464 = -4.464$ (<–3 → valid)", "Both solutions satisfy the domain condition.", "---", "### Final Answer", "The solutions to the equation $\log_2[(x + 3)(x – 1)] = 3$ are:\n[\nx = -1 + 2\sqrt{3} \quad \ ext{and} \quad x = -1 - 2\sqrt{3}\n]", "---", "### Why This Equation Matters", "Understanding equations like this builds foundational skills in logarithmic functions, domain restrictions, and algebraic manipulation. These concepts are essential for higher-level math including calculus, engineering problems, and computer science algorithms involving exponential growth and logarithmic scales.", "---", "### Tips for Learning and Teaching", "- Visualize the domain: Use number lines to emphasize where the logarithm is defined.\n- Practice conversion: Encourage converting logarithmic equations to exponentials before expanding.\n- Symbolic simplification emphasizes clean algebra, vital in avoiding errors.\n- Check every solution—a habit crucial for mastering real-world applications.", "---", "### Conclusion", "Solving $\log_2[(x + 3)(x - 1)] = 3$ elegantly combines logarithmic theory with algebraic techniques. Mastering this process strengthens analytical reasoning and prepares students for advanced mathematical challenges. Whether in the classroom or on your own, understanding step-by-step solving and domain awareness turns abstract equations into clear, solvable problems.", "---", "Keywords for SEO:\nlog base 2 logarithm equation, solve log₂((x+3)(x–1)) = 3, algebra equations, quadratic solutions, logarithmic domain, x+3 multiply x–1 equals 8, step-by-step logarithmic solving, precalculus logarithm problems", "Related searches:\n-how to solve log equations with product, convert log₂ to exponential form, logarithmic inequalities domain, solve log under product rule, quadratic from log base 2, applications of $\log_2$", "---", "Ready to practice? Try solving other log equations and verify every solution using domain rules!"]