Combine logs: log₂[(x + 3)(x − 1)] = 3

["SEO Article: Solving Combine Logarithms — Log₂[(x + 3)(x − 1)] = 3", "Unlocking the Solution to Logarithmic Equations: Combine Logs & Solve Log₂[(x + 3)(x − 1)] = 3", "Working with logarithmic equations is a key skill in algebra, especially when combining and solving them. One common exercise students face is solving equations involving logarithms with products inside the log, like Log₂[(x + 3)(x − 1)] = 3. Understanding how to combine logs and isolate variables is essential for mastering logarithmic problem-solving.", "In this article, we’ll break down step-by-step how to solve the equation Log₂[(x + 3)(x − 1)] = 3, explain the logic behind combining logarithmic expressions, and provide practical tips for tackling similar logarithmic equations.", "---", "### Step 1: Understanding the Logarithmic Equation\nThe given equation is:\n[\n\log_2[(x + 3)(x - 1)] = 3\n]\nThis expresses that the base-2 logarithm of the product ((x + 3)(x - 1)) equals 3. Recall that by definition:\n[\n\log_b(A) = C \implies A = b^C\n]\nApplying this here:\n[\n(x + 3)(x - 1) = 2^3 = 8\n]\nNow we convert the logarithmic equation into a standard algebraic form.", "---", "### Step 2: Expand and Rearrange\nFirst, expand the product:\n[\n(x + 3)(x - 1) = x^2 + 2x - 3\n]\nSo:\n[\nx^2 + 2x - 3 = 8\n]\nSubtract 8 from both sides:\n[\nx^2 + 2x - 11 = 0\n]\nNow solve this quadratic equation using the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nwhere (a = 1), (b = 2), (c = -11):\n[\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-11)}}{2} = \frac{-2 \pm \sqrt{4 + 44}}{2} = \frac{-2 \pm \sqrt{48}}{2}\n]\nSimplify (\sqrt{48} = 4\sqrt{3}):\n[\nx = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3}\n]", "---", "### Step 3: Check for Valid Solutions\nLogarithmic functions are only defined when their argument is positive. So, we must ensure:\n[\n(x + 3)(x - 1) > 0\n]\nSolve the inequality:\n- Roots of the expression: (x = -3) and (x = 1)\n- Test intervals:\n - For (x < -3): Test (x = -4): ((-1)(-5) = 5 > 0) ✅\n - For (-3 < x < 1): Test (x = 0): ((3)(-1) = -3 < 0) ❌\n - For (x > 1): Test (x = 2): ((5)(1) = 5 > 0) ✅", "Thus, valid solutions must satisfy (x < -3) or (x > 1).\nNow check the solutions:\n- (x = -1 + 2\sqrt{3} \approx -1 + 3.464 = 2.464 > 1) ✅\n- (x = -1 - 2\sqrt{3} \approx -1 - 3.464 = -4.464 < -3) ✅", "Both roots lie in valid domains.", "---", "### Step 4: Log Combining in Context\nThough we combined using the logarithmic identity (\log_b(A \cdot C) = \log_b A + \log_b C), direct solution here relied on exponentiation rather than log properties. However, understanding log combine rules helps when simplifying expressions:\n[\n\log_b[(x + 3)(x - 1)] = \log_b(x + 3) + \log_b(x - 1)\n]\nBut in this problem, combining was implicit via exponentiation:\n[\n(x+3)(x-1) = 8 \Rightarrow \log_2[(x+3)(x-1)] = \log_2 8 = 3\n]\nSo, combining logic and exponentiation ensures clarity and correctness.", "---", "### Final Answer\nThe solutions to (\log_2[(x + 3)(x - 1)] = 3) are:\n[\nx = -1 + 2\sqrt{3} \quad \ ext{and} \quad x = -1 - 2\sqrt{3}\n]\nBoth are valid due to domain restrictions.", "---", "### Tips for Solving Combine Log Problems\n- Use (b^C = A) to convert log equations to algebraic form\n- Expand and rearrange into standard polynomial or event inequality forms\n- Check solution validity within the domain of the logarithm’s argument\n- Understand when log add/divide/combine rules apply after simplification\n- Practice identifying valid solution intervals early", "---", "Mastering logarithmic equations starts with understanding logarithmic identities and domain constraints. With practice, solving combine logs—such as (\log_2[(x + 3)(x - 1)] = 3)—becomes intuitive and straightforward.", "Keywords: combine logs equation, solve log equations, logarithmic equations, log base 2, quadratic logarithms, algebra tutorial, solve log equations, logarithmic identity, domain of logs", "---", "Want more? Explore step-by-step logarithmic problem solvers or practice sheets on combining and solving log equations today!"]