\[ \log_2[x(x - 2)] = 3 \]

["# Solve the Equation: log₂[x(x – 2)] = 3", "Understanding logarithmic equations is essential for mastering algebra and preparing for advanced mathematics. One common challenge is solving equations like [ \log_2[x(x - 2)] = 3 ]. In this comprehensive guide, we’ll break down how to solve this logarithmic equation step-by-step, explain its key features, and offer practical tips to help you confidently work with logarithms in future problem-solving.", "## What Does log₂[x(x – 2)] = 3 Mean?", "The equation [ \log_2[x(x - 2)] = 3 ] asks: to what power must 2 divide to get the expression inside the logarithm, [x(x – 2)]?\nIn mathematical terms, logarithms convert exponential relationships into linear ones, making it easier to solve for the unknown variable.", "---", "## Step-by-Step Solution", "### Step 1: Eliminate the Logarithm", "To remove the base-2 logarithm, rewrite the equation in exponential form:", "[\nx(x - 2) = 2^3\n]", "Since (2^3 = 8), the equation becomes:", "[\nx(x - 2) = 8\n]", "### Step 2: Expand and Rearrange into Quadratic Form", "Expand the left-hand side:", "[\nx^2 - 2x = 8\n]", "Bring all terms to one side to form a standard quadratic equation:", "[\nx^2 - 2x - 8 = 0\n]", "### Step 3: Solve the Quadratic Equation", "Factor the quadratic:", "[\n(x - 4)(x + 2) = 0\n]", "Set each factor equal to zero:", "[\nx - 4 = 0 \quad \Rightarrow \quad x = 4\n]\n[\nx + 2 = 0 \quad \Rightarrow \quad x = -2\n]", "---", "## Validity Check: Domain of the Logarithm", "Before accepting both solutions, check whether they satisfy the original logarithmic expression [ \log_2[x(x - 2)] ]. The argument of a logarithm must be positive:", "[\nx(x - 2) > 0\n]", "Test each solution:", "- For (x = 4):\n (4(4 - 2) = 4 \ imes 2 = 8 > 0) → valid\n- For (x = -2):\n (-2(-2 - 2) = (-2)(-4) = 8 > 0) → valid", "Both values yield a positive argument, so both are mathematically valid.", "---", "## Final Answer", "[\n\boxed{x = 4 \quad \ ext{and} \quad x = -2}\n]", "These are the complete solutions to [ \log_2[x(x - 2)] = 3 ].", "---", "## Why This Equation Matters", "Solving equations like [ \log_2[x(x - 2)] = 3 ] builds critical thinking skills needed in calculus, computer science (e.g., binary systems), engineering, and finance—especially when working with logarithmic scales or exponential growth models.", "---", "## Key Takeaways", "- Use the definition of logarithms to convert (\log_b{A} = C) to (A = b^C).\n- Always verify domain restrictions to avoid invalid logarithmic expressions.\n- Factoring quadratic equations arising from logarithmic equations helps find exact solutions quickly.", "---", "## Practice Problems", "Try solving these similar equations:", "1. [ \log_3(x(x + 1)) = 2 ]\n2. [ \log_5[2x^2 - 7x + 3] = 1 ]", "---", "## Additional Resources", "- Learn more about logarithmic properties at Math is Fun – Logarithms\n- Watch step-by-step video tutorials on solving logarithmic equations via Khan Academy\n- Practice with interactive graphing tools like Desmos to visualize solutions", "---", "If you found this guide helpful, subscribe for more math tutorials and deep dives into solving logarithmic and exponential equations!", "---", "Keywords: logarithmic equation, solve log base 2, log₂[x(x – 2)] = 3, algebra steps, logarithmic domain, quadratic equation, exponential form, mathematical problem solving, math tutorial."]