Si \( \log_2(x) + \log_2(x-4) = 3 \), encuentra \( x \).

["How to Solve the Equation ( \log_2(x) + \log_2(x - 4) = 3 ): Step-by-Step Guide", "Solving logarithmic equations can seem challenging at first, but with the right approach, equations like ( \log_2(x) + \log_2(x - 4) = 3 ) become manageable. In this article, we’ll explore how to find all valid values of ( x ) that satisfy this equation, using key logarithm properties and algebraic steps.", "---", "### Understanding the Equation", "We begin with:\n[\n\log_2(x) + \log_2(x - 4) = 3\n]", "This equation involves the sum of two logarithms with the same base, base 2. A key logarithmic property states:\n[\n\log_b(a) + \log_b(c) = \log_b(a \cdot c)\n]\nProvided that ( a > 0 ) and ( c > 0 ), so their product is positive.", "---", "### Step 1: Combine the Logarithms", "Apply the logarithm sum rule:\n[\n\log_2(x) + \log_2(x - 4) = \log_2\left(x(x - 4)\right)\n]", "Substitute back into the equation:\n[\n\log_2\left(x(x - 4)\right) = 3\n]", "---", "### Step 2: Remove the Logarithm", "To eliminate the logarithm, recall that if ( \log_b(y) = z ), then ( y = b^z ). Here, ( b = 2 ), ( y = x(x - 4) ), and ( z = 3 ):\n[\nx(x - 4) = 2^3\n]\n[\nx(x - 4) = 8\n]", "---", "### Step 3: Expand and Rearrange into a Quadratic Equation", "Expand the left side:\n[\nx^2 - 4x = 8\n]\nBring all terms to one side:\n[\nx^2 - 4x - 8 = 0\n]", "---", "### Step 4: Solve the Quadratic Equation", "Use the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nFor ( a = 1, b = -4, c = -8 ):\n[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-8)}}{2(1)} = \frac{4 \pm \sqrt{16 + 32}}{2} = \frac{4 \pm \sqrt{48}}{2}\n]", "Simplify ( \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} ):\n[\nx = \frac{4 \pm 4\sqrt{3}}{2} = 2 \pm 2\sqrt{3}\n]", "So the two potential solutions are:\n[\nx = 2 + 2\sqrt{3} \quad \ ext{and} \quad x = 2 - 2\sqrt{3}\n]", "---", "### Step 5: Check Validity Using the Domain of the Logarithm", "Recall that logarithms are only defined for positive arguments. So both ( x > 0 ) and ( x - 4 > 0 ) must hold.", "- ( x > 0 ):\n - ( 2 + 2\sqrt{3} \approx 2 + 3.464 = 5.464 > 0 ) — valid\n - ( 2 - 2\sqrt{3} \approx 2 - 3.464 = -1.464 ) — not valid (negative)", "- ( x - 4 > 0 \Rightarrow x > 4 ):\n - ( 2 + 2\sqrt{3} \approx 5.464 > 4 ) — valid\n - ( 2 - 2\sqrt{3} \approx -1.464 ) — invalid", "Only ( x = 2 + 2\sqrt{3} ) satisfies both conditions.", "---", "### Final Answer", "The only valid solution is:\n[\n\boxed{x = 2 + 2\sqrt{3}}\n]", "---", "### Summary", "To solve ( \log_2(x) + \log_2(x - 4) = 3 ):\n1. Use logarithmic product rule to combine logs.\n2. Turn the equation into an exponential form.\n3. Rearrange into a quadratic equation.\n4. Solve using the quadratic formula.\n5. Verify solutions against the logarithmic domain.", "Using these steps ensures correct and efficient problem-solving in logarithmic equations.", "---", "Keywords: logarithmic equation, solve ( \log_2(x) + \log_2(x - 4) = 3 ), step-by-step, logarithm properties, algebraic solution, domain of logarithm, quadratic formula."]