If \(\log_2(x) + \log_2(x-3) = 3\), what is x?

["# Solving the Logarithmic Equation: If (\log_2(x) + \log_2(x-3) = 3), What Is (x)?", "Understanding how to solve equations involving logarithms is essential in many areas of mathematics and real-world applications, such as finance, engineering, and computer science. One common type of problem is when logarithmic expressions are combined using properties of logarithms. In this article, we’ll explore how to solve the equation:", "[\n\log_2(x) + \log_2(x - 3) = 3\n]", "and find the value of (x).", "---", "## Step 1: Combine the Logarithmic Expressions", "Using the logarithmic identity that states:", "[\n\log_b(A) + \log_b(B) = \log_b(AB)\n]", "we can combine the left-hand side:", "[\n\log_2(x) + \log_2(x - 3) = \log_2(x(x - 3)) = 3\n]", "So the equation becomes:", "[\n\log_2(x(x - 3)) = 3\n]", "---", "## Step 2: Eliminate the Logarithm Using Exponentiation", "To remove the log, we rewrite the equation in exponential form. Since the base is 2:", "[\nx(x - 3) = 2^3\n]", "[\nx(x - 3) = 8\n]", "---", "## Step 3: Expand, Rearrange, and Solve the Quadratic Equation", "Expand the left-hand side:", "[\nx^2 - 3x = 8\n]", "Bring all terms to one side to form a standard quadratic equation:", "[\nx^2 - 3x - 8 = 0\n]", "Solve using the quadratic formula:", "[\nx = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-8)}}{2(1)} = \frac{3 \pm \sqrt{9 + 32}}{2} = \frac{3 \pm \sqrt{41}}{2}\n]", "Thus, the two possible solutions are:", "[\nx = \frac{3 + \sqrt{41}}{2} \quad \ ext{and} \quad x = \frac{3 - \sqrt{41}}{2}\n]", "---", "## Step 4: Check for Validity in the Original Equation", "Logarithmic functions are only defined for positive arguments. Therefore, both (x > 0) and (x - 3 > 0) must hold, meaning:", "[\nx > 3\n]", "Now evaluate the solutions:", "- (\frac{3 + \sqrt{41}}{2}): Since (\sqrt{41} \approx 6.4), this gives approximately ((3 + 6.4)/2 = 4.7 > 3) — valid.\n- (\frac{3 - \sqrt{41}}{2}): This is approximately ((3 - 6.4)/2 = -1.7 < 3) — invalid, since logarithms are undefined.", "---", "## Step 5: Final Answer", "The only valid solution is:", "[\n\boxed{x = \frac{3 + \sqrt{41}}{2}}\n]", "---", "## Conclusion", "By applying logarithmic properties and solving the resulting quadratic equation—while carefully checking domain constraints—we efficiently found the value of (x) that satisfies the original equation. This step-by-step method is key to solving logarithmic equations accurately and effectively.", "For anyone studying logarithms, mastering these steps helps build strong problem-solving skills applicable to both academic and real-world challenges. If you're stuck on a logarithmic equation, remember to combine logs first, eliminate them with exponentiation, solve carefully, and always verify the domain!"]