If \( \log_2(x) + \log_2(x-1) = 3 \), find the value of \( x \).
["Solving the Logarithmic Equation: ( \log_2(x) + \log_2(x-1) = 3 )", "Understanding logarithmic equations is essential in algebra, and today we’ll solve a common but powerful type: equations involving log properties. This specific problem—If ( \log_2(x) + \log_2(x-1) = 3 ), find the value of ( x )—can be tackled efficiently using logarithmic identities and algebraic manipulation.", "---", "### Understanding the Equation", "The equation is:\n[\n\log_2(x) + \log_2(x - 1) = 3\n]\nBoth terms are logarithms with base 2. A key logarithmic property allows us to combine sums into a product:\n[\n\log_b(A) + \log_b(B) = \log_b(AB)\n]\nApplying this identity to our equation:\n[\n\log_2(x(x - 1)) = 3\n]\n[\n\log_2(x^2 - x) = 3\n]", "---", "### Eliminate the Logarithm", "To solve for ( x ), rewrite the logarithmic equation in its exponential form. Since the base is 2:\n[\nx^2 - x = 2^3\n]\n[\nx^2 - x = 8\n]\nRearranging gives a standard quadratic equation:\n[\nx^2 - x - 8 = 0\n]", "---", "### Solving the Quadratic Equation", "Use the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nHere, ( a = 1 ), ( b = -1 ), and ( c = -8 ):\n[\nx = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-8)}}{2(1)} = \frac{1 \pm \sqrt{1 + 32}}{2} = \frac{1 \pm \sqrt{33}}{2}\n]", "This yields two potential solutions:\n[\nx = \frac{1 + \sqrt{33}}{2} \quad \ ext{and} \quad x = \frac{1 - \sqrt{33}}{2}\n]", "---", "### Applying Domain Restrictions", "Since logarithms are only defined for positive arguments, the original equation ( \log_2(x) + \log_2(x - 1) = 3 ) requires:\n- ( x > 0 )\n- ( x - 1 > 0 \Rightarrow x > 1 )", "Thus, ( x ) must be greater than 1. Now, examine both solutions:\n- ( \frac{1 + \sqrt{33}}{2} \approx \frac{1 + 5.74}{2} = \frac{6.74}{2} \approx 3.37 > 1 ) ✅ valid\n- ( \frac{1 - \sqrt{33}}{2} \approx \frac{1 - 5.74}{2} = \frac{-4.74}{2} \approx -2.37 < 1 ) ❌ invalid", "The negative root fails the domain condition.", "---", "### Final Answer", "The only valid solution is:\n[\nx = \frac{1 + \sqrt{33}}{2}\n]", "Verifying this value in the original equation confirms it satisfies ( \log_2(x) + \log_2(x-1) = 3 ).", "---", "### Why This Problem Matters", "This example illustrates how logarithmic equations leverage powerful properties (like the product rule) and combine them with algebra to find exact solutions—skills essential for advanced math, engineering, and science applications.", "---", "Key takeaways:\n- Use logarithmic identities to simplify expressions.\n- Always check domain restrictions before accepting solutions.\n- Quadratic formulas are powerful tools for solving equations involving logs.", "Start sharp—master these steps, and you’ll tackle logarithmic puzzles with confidence!", "---\nIf you found this guide helpful, consider exploring logarithmic equations with different bases or combining multiple log properties next!*"]
