If \( \log_2(x) + \log_2(x - 2) = 3 \), what is the value of \( x \)?

["Understanding and Solving the Logarithmic Equation: If ( \log_2(x) + \log_2(x - 2) = 3 ), What is the Value of ( x )?", "Logarithmic equations often spark curiosity and challenge students and math enthusiasts alike. Solving ( \log_2(x) + \log_2(x - 2) = 3 ) is a great example of how logarithmic properties simplify complex expressions—and how to find the correct value of ( x ) using careful steps.", "### Step 1: Use Logarithmic Properties to Combine Terms", "We begin by applying the sum rule of logarithms, which states that:\n[\n\log_b(m) + \log_b(n) = \log_b(m \cdot n)\n]\nApplying this rule to the equation:\n[\n\log_2(x) + \log_2(x - 2) = \log_2(x(x - 2))\n]\nSo the equation becomes:\n[\n\log_2(x(x - 2)) = 3\n]", "### Step 2: Convert Logarithmic Equation to Exponential Form", "Recall that ( \log_b(A) = C ) is equivalent to ( A = b^C ). Applying this:\n[\nx(x - 2) = 2^3 = 8\n]", "### Step 3: Expand and Solve the Quadratic Equation", "Expanding the left-hand side:\n[\nx^2 - 2x = 8\n]\nBring all terms to one side:\n[\nx^2 - 2x - 8 = 0\n]\nNow solve using factoring:\n[\n(x - 4)(x + 2) = 0\n]\nSo the potential solutions are:\n[\nx = 4 \quad \ ext{or} \quad x = -2\n]", "### Step 4: Check for Valid Solutions in the Original Equation", "Before accepting a solution, always verify it in the original logarithmic expression to ensure we’re working within the domain of logarithms.", "Logarithms are only defined for positive arguments, so:", "- For ( \log_2(x) ): ( x > 0 )\n- For ( \log_2(x - 2) ): ( x - 2 > 0 \Rightarrow x > 2 )", "Now check:\n- ( x = 4 ):\n ( \log_2(4) ) and ( \log_2(2) ) are both defined. ✅\n- ( x = -2 ):\n ( \log_2(-2) ) is undefined. ❌", "### Final Answer", "Only ( x = 4 ) satisfies the domain requirements and solves the equation.", "Therefore, the value of ( x ) is ( 4 ).", "---", "### Why This Equation Matters", "Solving equations involving sum of logs reinforces key algebraic skills such as:\n- Applying logarithmic identities\n- Manipulating and solving quadratic equations\n- Validating solutions in the context of the original domain", "Understanding these steps helps build confidence in handling logarithmic expressions—essential in advanced mathematics, engineering, and scientific computations.", "If you're working on logarithmic equations, always:\n1. Ensure arguments are positive\n2. Combine logs where possible\n3. Verify solutions\nFollowing these rules ensures accuracy and deeper insight.", "---", "Keywords: ( \log_2(x) + \log_2(x - 2) = 3 ), solve logarithmic equation, value of ( x ), logarithmic properties, quadratic equation, domain of logarithms, steps to solve, math problem solutions.\nMeta Description: Learn how to solve ( \log_2(x) + \log_2(x - 2) = 3 ), find ( x = 4 ) as the valid solution, and understand logarithmic domain and verification steps."]