If \( \log_2(x) + \log_2(x - 3) = 2 \), find the value of \( x \).

If \( \log_2(x) + \log_2(x - 3) = 2 \), find the value of \( x \).

["Solving the Logarithmic Equation: Find ( x ) from ( \log_2(x) + \log_2(x - 3) = 2 )", "Understanding how to solve logarithmic equations is essential in algebra and higher-level math, especially in fields like engineering, computer science, and applied sciences. Today, we explore a classic problem: If ( \log_2(x) + \log_2(x - 3) = 2 ), find the value of ( x ).", "---", "### Step 1: Use Logarithm Properties to Combine Terms", "One of the key properties of logarithms is:\n[\n\log_b(A) + \log_b(B) = \log_b(A \cdot B)\n]\nApplying this property to the left-hand side of the equation:", "[\n\log_2(x) + \log_2(x - 3) = \log_2(x(x - 3))\n]", "So the equation becomes:", "[\n\log_2(x(x - 3)) = 2\n]", "---", "### Step 2: Convert Logarithmic Equation to Exponential Form", "Recall that ( \log_b(A) = C ) implies ( A = b^C ).\nHere, ( b = 2 ), ( C = 2 ), so:", "[\nx(x - 3) = 2^2 = 4\n]", "---", "### Step 3: Simplify and Solve the Quadratic Equation", "Expanding the left-hand side:", "[\nx^2 - 3x = 4\n\Rightarrow x^2 - 3x - 4 = 0\n]", "This is a standard quadratic equation. Solve using factoring:", "[\nx^2 - 3x - 4 = (x - 4)(x + 1) = 0\n]", "So the potential solutions are:", "[\nx = 4 \quad \ ext{or} \quad x = -1\n]", "---", "### Step 4: Check for Validity Using Domain Restrictions", "Since logarithms are only defined for positive real numbers, both ( \log_2(x) ) and ( \log_2(x - 3) ) require:", "[\nx > 0 \quad \ ext{and} \quad x - 3 > 0 \Rightarrow x > 3\n]", "Thus, ( x = -1 ) is not valid because it violates ( x > 3 ).\nOnly ( x = 4 ) satisfies the domain condition.", "---", "### Step 5: Verify the Solution", "Plug ( x = 4 ) into the original equation:", "[\n\log_2(4) + \log_2(4 - 3) = \log_2(4) + \log_2(1) = 2 + 0 = 2\n]", "This checks out correctly.", "---", "### Conclusion", "The solution to the equation ( \log_2(x) + \log_2(x - 3) = 2 ) is:", "[\n\boxed{4}\n]", "Mastering logarithmic equations like this enhances problem-solving skills and forms the foundation for tackling real-world scientific and engineering challenges. Always remember to check the domain—your solution must make sense within the logarithm’s valid input range.", "---", "Keywords:\nlogarithmic equation solution, solve ( \log_2(x) + \log_2(x - 3) = 2 ), step-by-step math, how to solve log equations, find ( x ) for logarithmic logs, algebra techniques, quadratic from log log equation, domain of logarithms, logarithmic properties.", "Meta Description:\nLearn how to solve ( \log_2(x) + \log_2(x - 3) = 2 ) using log properties, quadratic solving, and domain checks. Step-by-step explanation with verification."]

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