If \( \log_2(x) + \log_2(x - 4) = 3 \), find \( x \).
["Solving the Equation: ( \log_2(x) + \log_2(x - 4) = 3 )", "Understanding logarithmic equations can seem challenging at first, but with the right steps and properties, solving equations like ( \log_2(x) + \log_2(x - 4) = 3 ) becomes straightforward. This article walks you through solving this logarithmic equation to find the value(s) of ( x ).", "---", "### Step 1: Combine Logarithms Using Product Rule", "The equation involves the sum of two logarithms with the same base:", "[\n\log_2(x) + \log_2(x - 4) = 3\n]", "Using the logarithmic product rule, ( \log_b(a) + \log_b(c) = \log_b(a \cdot c) ), you can combine the terms:", "[\n\log_2(x(x - 4)) = 3\n]", "Simplify the expression inside the log:", "[\n\log_2(x^2 - 4x) = 3\n]", "---", "### Step 2: Convert the Logarithmic Equation to Exponential Form", "To eliminate the logarithm, recall the definition:", "[\n\log_b(N) = C \quad \ ext{is equivalent to} \quad N = b^C\n]", "Applying this property:", "[\nx^2 - 4x = 2^3\n]", "[\nx^2 - 4x = 8\n]", "---", "### Step 3: Rearrange into a Standard Quadratic Equation", "Move all terms to one side to form a standard quadratic:", "[\nx^2 - 4x - 8 = 0\n]", "---", "### Step 4: Solve the Quadratic Equation", "Use the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) with ( a = 1 ), ( b = -4 ), and ( c = -8 ):", "[\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} ):", "[\n\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}\n]", "Thus,", "[\nx = \frac{4 \pm 4\sqrt{3}}{2} = 2 \pm 2\sqrt{3}\n]", "---", "### Step 5: Determine the Valid Solution(s) Using Domain Rules", "Since logarithms are only defined for positive arguments, check the domain of the original equation:", "[\n\log_2(x) \quad \ ext{requires} \quad x > 0\n]\n[\n\log_2(x - 4) \quad \ ext{requires} \quad x - 4 > 0 \Rightarrow x > 4\n]", "So the domain is ( x > 4 ). Now evaluate both potential solutions:", "- ( x = 2 + 2\sqrt{3} \approx 2 + 3.464 = 5.464 ) → valid (greater than 4)\n- ( x = 2 - 2\sqrt{3} \approx 2 - 3.464 = -1.464 ) → not valid (less than 4)", "---", "### Final Answer", "The only valid solution is:", "[\nx = 2 + 2\sqrt{3}\n]", "---", "### Conclusion", "Solving ( \log_2(x) + \log_2(x - 4) = 3 ) involves combining logs, converting to exponential form, solving the resulting quadratic, and carefully checking domain restrictions. Remember: always verify solutions in the original equation when working with logarithmic functions—especially ensuring the arguments remain positive.", "Understanding this method empowers you to solve similar logarithmic equations confidently.", "---", "Keywords:\n( \log_2(x) + \log_2(x - 4) = 3 ), solve logarithmic equation, logarithmic problems, linearize log equation, solve quadratic from logs, domain of logarithms, step-by-step logarithmic equation, mathematical problem solving.", "---", "Related Articles:\n- How to Simplify Logarithmic Expressions\n- Step-by-Step Guide to Solving Logarithmic Equations\n- Common Errors in Logarithmic Equations and How to Avoid Them"]
