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

["Understanding Combine Logs: Solving (\log_2(x(x - 3)) = 3)", "Logarithmic equations are powerful tools in algebra, helping to simplify complex expressions involving growth, decay, and scaling—key concepts in science, finance, and engineering. One common type involves solving equations written in logarithmic form with base 2. In this guide, we’ll explore how to solve the equation (\log_2(x(x - 3)) = 3) step-by-step using combine logs techniques, making it easier to understand and apply.", "---", "### What Does (\log_2(x(x - 3)) = 3) Mean?", "The equation (\log_2(x(x - 3)) = 3) asks: For which values of (x) is the logarithm (base 2) of (x(x - 3)) equal to 3?\nLogarithmic equations convert multiplicative relationships into additive ones, making them simpler to solve—especially when combined with log properties.", "---", "### Step 1: Combine Logs Using Log Product Rule", "The key to solving this equation lies in recalling the logarithm product rule:", "[\n\log_b(M \cdot N) = \log_b M + \log_b N\n]", "Apply this to the left side of the equation:", "[\n\log_2(x(x - 3)) = \log_2 x + \log_2(x - 3)\n]", "Rewriting the original equation gives:", "[\n\log_2 x + \log_2 (x - 3) = 3\n]", "---", "### Step 2: Convert to Exponential Form (Optional for Clarity)", "Though not strictly necessary, converting back can reinforce understanding. Recall that:", "[\n\log_b Y = z \quad \ ext{implies} \quad Y = b^z\n]", "So:", "[\nx(x - 3) = 2^3 = 8\n]", "This step confirms the earlier logarithmic equation is equivalent to:", "[\nx(x - 3) = 8\n]", "---", "### Step 3: Expand and Rearrange into Quadratic Form", "Expand the left-hand side:", "[\nx^2 - 3x = 8\n]", "Bring all terms to one side:", "[\nx^2 - 3x - 8 = 0\n]", "---", "### Step 4: Solve Using the Quadratic Formula", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, (a = 1), (b = -3), (c = -8):", "[\nx = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-8)}}{2(1)} = \frac{3 \pm \sqrt{9 + 32}}{2} = \frac{3 \pm \sqrt{41}}{2}\n]", "So the two solutions are:", "[\nx = \frac{3 + \sqrt{41}}{2} \quad \ ext{and} \quad x = \frac{3 - \sqrt{41}}{2}\n]", "---", "### Step 5: Check for Domain Validity", "Since we’re dealing with a logarithm base 2 of (x(x - 3)), the argument must be positive:", "[\nx(x - 3) > 0\n]", "Solve the inequality:", "- The roots of (x(x - 3) = 0) are (x = 0) and (x = 3).\n- The parabola opens up, so the product is positive when (x < 0) or (x > 3).", "Now examine both solutions:", "- (\frac{3 + \sqrt{41}}{2} \approx \frac{3 + 6.4}{2} = 4.7 > 3) → valid.\n- (\frac{3 - \sqrt{41}}{2} \approx \frac{3 - 6.4}{2} = -1.7 < 0) → invalid, since (x(x-3) < 0).", "Thus, the only valid solution is:", "[\nx = \frac{3 + \sqrt{41}}{2}\n]", "---", "### Why This Matters: Summary of Key Logic", "- Combine logs using the product rule to simplify: (\log_b(x(x - 3)) = \log_b x + \log_b (x - 3)).\n- Convert if needed, but underlying algebra remains valid.\n- Always check domain restrictions—logarithms demand positive arguments.\n- Solve algebraically and validate solutions against constraints.", "---", "### Real-World Applications", "Equations like (\log_2(x(x - 3)) = 3) appear in modeling exponential growth, signal processing, and information theory, where scaling and logarithmic relationships describe signal strength, data entropy, or population dynamics.", "---", "### Final Takeaway", "Mastering combine logs and careful domain analysis transforms logarithmic equations from intimidating puzzles into solvable, meaningful problems. Whether you’re solving for (x) in a classroom, coding an algorithm, or analyzing scientific data, understanding this workflow builds strong algebraic intuition.", "---", "Keywords for SEO:\n(\log_2(x(x - 3)) = 3), solving logarithmic equations, combine logs technique, logarithmic product rule, domain of log function, quadratic formula application", "Meta Description:\nLearn how to solve (\log_2(x(x - 3)) = 3) using log combination rules, expand to quadratic form, and validate solutions. Step-by-step explanation with real-world context and domain checks."]