Combine logs: \( \log_2(x(x - 2)) = 3 \) → \( x(x - 2) = 2^3 = 8 \).
["# Solving Combine Log Equations: Understanding ( \log_2(x(x - 2)) = 3 )", "Logarithmic equations often appear challenging at first, but understanding the step-by-step process can make them straightforward. In this article, we’ll dive into solving the equation:", "[\n\log_2(x(x - 2)) = 3\n]", "and explain how to combine and simplify the logs to isolate ( x ).", "---", "## Why Logarithms Matter in Algebra", "The logarithmic form ( \log_b A = C ) translates directly into exponential form:", "[\nA = b^C\n]", "This key conversion is essential for solving equations involving logarithms like the one above.", "---", "## Step-by-Step Guide to Solve ( \log_2(x(x - 2)) = 3 )", "### Step 1: Convert to Exponential Form", "Since the base of the log is 2, rewrite the equation as:", "[\nx(x - 2) = 2^3\n]", "Because ( \log_2(A) = 3 ) means ( A = 2^3 ).", "### Step 2: Simplify the Exponent", "Compute ( 2^3 ):", "[\nx(x - 2) = 8\n]", "---", "## Expanding and Rearranging", "### Step 3: Expand the Quadratic Expression", "Multiply the left side:", "[\nx^2 - 2x = 8\n]", "### Step 4: Move All Terms to One Side", "Bring 8 to the left for standard quadratic form:", "[\nx^2 - 2x - 8 = 0\n]", "---", "## Solve the Quadratic Equation", "### Step 5: Use Factoring or the Quadratic Formula", "Try factoring:", "[\nx^2 - 2x - 8 = (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]", "---", "## Step 6: Check Validity—Domain Matters", "Recall the original log expression is ( \log_2(x(x - 2)) ), which is only defined when:", "[\nx(x - 2) > 0\n]", "Analyze the inequality ( x(x - 2) > 0 ):", "- The roots are ( x = 0 ) and ( x = 2 )\n- The expression is positive when ( x < 0 ) or ( x > 2 )", "Now check both solutions:", "- ( x = 4 ): ( 4 > 2 ), valid\n- ( x = -2 ): ( -2 < 0 ), valid", "Both values satisfy the domain condition.", "---", "## Final Answer", "The solutions to ( \log_2(x(x - 2)) = 3 ) are:", "[\n\boxed{x = 4 \quad \ ext{and} \quad x = -2}\n]", "---", "## Why This Process Works", "- Converting logs to exponentials eliminates the log function, turning a logarithmic equation into a solvable algebraic form.\n- Expanding and rearranging allows standard quadratic techniques.\n- Checking domain ensures solutions are valid within the original logarithmic constraints.", "Understanding this process builds confidence in solving more complex log equations and strengthens algebraic problem-solving skills.", "---", "### Related Keywords for SEO:", "- Solve logarithmic equations\n- How to solve ( \log_b(x(x - 2)) = 3 )\n- Convert log to exponential form\n- Solve quadratic equations from logs\n- Domain of logarithmic functions\n- Step-by-step logarithm solving guide\n- Step-by-step quadratic equation solution", "---", "If you found this guide helpful, explore more on exponential models, logarithmic identities, and algebraic manipulation techniques!"]
