Solve for \(x\): \(\log_2(x^2 - 5x) = 3\).

["# Solve for (x): (\log_2(x^2 - 5x) = 3)", "Understanding logarithmic equations is essential for mastering algebra and solving real-world problems in science, engineering, and finance. One common challenge is solving equations of the form (\log_b(f(x)) = c), where logarithms are used to compute exponential relationships. In this article, we’ll find the solution to the logarithmic equation:\n[\n\log_2(x^2 - 5x) = 3\n]", "---", "## Step 1: Eliminate the Logarithm", "To solve for (x), begin by converting the logarithmic equation into its equivalent exponential form. For base 2, this means:", "[\nx^2 - 5x = 2^3\n]", "Since (2^3 = 8), the equation simplifies to:", "[\nx^2 - 5x = 8\n]", "---", "## Step 2: Rearrange into Standard Quadratic Form", "Move all terms to one side to form a standard quadratic equation:", "[\nx^2 - 5x - 8 = 0\n]", "This is now in the form (ax^2 + bx + c = 0), suitable for applying the quadratic formula.", "---", "## Step 3: Apply the Quadratic Formula", "The quadratic formula is:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nHere, (a = 1), (b = -5), and (c = -8).", "Plugging in the values:", "[\nx = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-8)}}{2(1)} = \frac{5 \pm \sqrt{25 + 32}}{2} = \frac{5 \pm \sqrt{57}}{2}\n]", "---", "## Step 4: Consider Domain Restrictions", "Before accepting the solutions, recall that logarithmic functions are only defined for positive arguments. The original equation involves:", "[\n\log_2(x^2 - 5x)\n]", "Thus, we require:", "[\nx^2 - 5x > 0\n]", "Solve the inequality (x^2 - 5x > 0):", "Factor:", "[\nx(x - 5) > 0\n]", "The critical points are (x = 0) and (x = 5). Testing intervals:", "- For (x < 0): Positive (e.g., (x = -1 \Rightarrow (-1)(-6) = 6 > 0))\n- For (0 < x < 5): Negative (e.g., (x = 2 \Rightarrow 2(-3) = -6 < 0))\n- For (x > 5): Positive (e.g., (x = 6 \Rightarrow 6(1) = 6 > 0))", "So the domain is (x < 0) or (x > 5).", "---", "## Step 5: Check Solutions Against the Domain", "We found:", "[\nx = \frac{5 + \sqrt{57}}{2} \quad \ ext{and} \quad x = \frac{5 - \sqrt{57}}{2}\n]", "Approximate (\sqrt{57} \approx 7.55), so:", "- (x_1 = \frac{5 + 7.55}{2} \approx 6.275 > 5) → valid\n- (x_2 = \frac{5 - 7.55}{2} \approx -1.275 < 0) → valid", "Both solutions satisfy (x < 0) or (x > 5), so both lie in the domain of the original logarithmic expression.", "---", "## Final Answer", "The solutions to the equation (\log_2(x^2 - 5x) = 3) are:", "[\nx = \frac{5 \pm \sqrt{57}}{2}\n]", "These two real values are both valid due to domain constraints. Solving logarithmic equations step-by-step—converting to exponential form, solving quadratics, and verifying domain restrictions—is crucial for accurate results in algebra and applied mathematics.", "---", "### Key Takeaways:", "- Always convert logarithmic equations to exponential form.\n- Solve resulting polynomials using appropriate formulas.\n- Never forget to check that solutions satisfy the original function's domain.\n- Accurate domain analysis prevents extraneous solutions.", "Mastering these steps empowers you to tackle a wide range of logarithmic problems confidently!", "---", "Keywords: solve for x logarithmic equation, logarithmic equation with base 2, quadratic equation (x^2 - 5x = 8), domain restriction in logarithms, step-by-step solution, algebra tutorial, logarithm and exponential conversion, how to solve (\log_2(x^2 - 5x) = 3)", "---", "Related Search Terms:\n- How to solve (\log_2(x^2 - 6x) = 2)?\n- Solve (\log_2(x^2 - 5x) = 3) step-by-step\n- Algebra: logarithmic equations with domain checks\n- Solve (\log(x^2 - 5x) = 3) (common variant with natural log notation)", "---", "Invest time in practicing these techniques—logarithmic equations are foundational for advanced math and science fields!"]