Problem:** Solve for \( x \): \( \log_2(x + 3) + \log_2(x - 1) = 3 \).
["Problem: Solve for ( x ): ( \log_2(x + 3) + \log_2(x - 1) = 3 )", "---", "### Simplify and Solve the Logarithmic Equation — A Step-by-Step Guide", "If you’ve ever encountered an equation involving logarithms like ( \log_2(x + 3) + \log_2(x - 1) = 3 ), you’re not alone — logarithmic equations can be challenging, but with the right approach, they become manageable. In this article, we’ll solve the equation ( \log_2(x + 3) + \log_2(x - 1) = 3 ) step-by-step, explain the key concepts, and emphasize the important rules you must remember to avoid common mistakes.", "---", "### Step 1: Use Logarithm Properties to Combine Terms", "Recall that the sum of logarithms with the same base can be combined using the product rule:", "[\n\log_b(A) + \log_b(B) = \log_b(AB)\n]", "Applying this to our equation:", "[\n\log_2(x + 3) + \log_2(x - 1) = \log_2\left( (x + 3)(x - 1) \right)\n]", "So the equation becomes:", "[\n\log_2\left( (x + 3)(x - 1) \right) = 3\n]", "---", "### Step 2: Eliminate the Logarithm Using Exponent Form", "Since the logarithm base is 2, rewrite the equation in exponential form:", "[\n(x + 3)(x - 1) = 2^3\n]", "[\n(x + 3)(x - 1) = 8\n]", "---", "### Step 3: Expand the Left Side", "Multiply the left-hand side:", "[\nx^2 - x + 3x - 3 = x^2 + 2x - 3\n]", "So:", "[\nx^2 + 2x - 3 = 8\n]", "---", "### Step 4: Bring All Terms to One Side to Form a Quadratic Equation", "Subtract 8 from both sides:", "[\nx^2 + 2x - 11 = 0\n]", "---", "### Step 5: 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 ), so:", "[\nx = \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} ):", "[\nx = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3}\n]", "So the two solutions are:", "[\nx = -1 + 2\sqrt{3} \quad \ ext{and} \quad x = -1 - 2\sqrt{3}\n]", "---", "### Step 6: Check for Extraneous Solutions", "because logarithms are only defined for positive arguments, we must ensure both solutions make ( \log_2(x + 3) ) and ( \log_2(x - 1) ) valid (i.e., both ( x + 3 > 0 ) and ( x - 1 > 0 )).", "- ( x = -1 + 2\sqrt{3} ):\n ( \sqrt{3} \approx 1.732 \Rightarrow 2\sqrt{3} \approx 3.464 )\n So ( x \approx -1 + 3.464 = 2.464 ), which satisfies ( x > 1 ) and ( x > -3 ), both valid.", "- ( x = -1 - 2\sqrt{3} \approx -1 - 3.464 = -4.464 ):\n This gives ( x - 1 < 0 ), so ( \log_2(x - 1) ) is undefined — extraneous.", "---", "### Final Valid Solution:", "[\n\boxed{x = -1 + 2\sqrt{3}}\n]", "---", "### Key Takeaways for Solving Logarithmic Equations", "- Combine logs using rules: ( \log_b A + \log_b B = \log_b(AB) )\n- Convert to exponential form: ( \log_b(A) = C \Rightarrow A = b^C )\n- Check domain restrictions: arguments of logarithms must be positive\n- Verify all potential solutions to avoid extraneous roots", "---", "### Why This Problem Matters", "Equations of the form ( \log_b(f(x)) = C ) appear frequently in science, engineering, and finance where logarithmic relationships model growth, decay, or ratios. Mastering these concepts builds a strong foundation for solving more complex equations and real-world modeling problems.", "---", "Keywords: solve ( \log_2(x + 3) + \log_2(x - 1) = 3 ), logarithmic equations, logarithm rules, step-by-step solution, domain checking, quadratic formula, math problems, algebra tutor", "---", "Learn how to solve logarithmic equations confidently and avoid common pitfalls — practice applying these steps and review logarithmic properties regularly!"]
