\[ \log_2(x^2 - 4) = 3 \Rightarrow x^2 - 4 = 2^3 = 8. \]

["Solving the Logarithmic Equation: ∣ log₂(x² – 4) = 3 ⇒ x² – 4 = 8 ⇒ Solve for x", "Learning how to solve logarithmic equations is essential for mastering algebra and logarithmic functions. One common yet insightful problem is solving equations in the form:", "[\n\log_2(x^2 - 4) = 3\n]", "This equation presents a perfect opportunity to explore the connection between logarithms and exponents. Let’s walk step-by-step through the solution and understand why ( x^2 - 4 = 2^3 = 8 ) leads us to the correct values of ( x ).", "---", "### Step 1: Understand the domain of the logarithm", "Before solving, always check the domain of the logarithmic expression. Since the logarithm ( \log_2(y) ) is defined only when ( y > 0 ), we require:", "[\nx^2 - 4 > 0\n]", "Solve this inequality:", "[\nx^2 > 4 \Rightarrow |x| > 2 \Rightarrow x < -2 \ ext{ or } x > 2\n]", "This domain restriction ensures any solution we find must satisfy this condition.", "---", "### Step 2: Convert the logarithmic equation to exponential form", "Given:", "[\n\log_2(x^2 - 4) = 3\n]", "By the definition of logarithms, this implies:", "[\nx^2 - 4 = 2^3 = 8\n]", "So we now solve:", "[\nx^2 - 4 = 8\n]", "---", "### Step 3: Solve the resulting quadratic equation", "Add 4 to both sides:", "[\nx^2 = 8 + 4 = 12\n]", "Now take the square root of both sides:", "[\nx = \pm \sqrt{12} = \pm 2\sqrt{3}\n]", "---", "### Step 4: Verify solutions against the domain condition", "Recall the domain requirement: ( x < -2 ) or ( x > 2 ).\nCheck:\n- ( 2\sqrt{3} \approx 3.464 ), which is greater than 2 — valid.\n- ( -2\sqrt{3} \approx -3.464 ), which is less than -2 — also valid.", "Both solutions lie in the valid domain.", "---", "### Final Answer and Summary", "The solutions to the equation ( \log_2(x^2 - 4) = 3 ) are:", "[\nx = 2\sqrt{3} \quad \ ext{or} \quad x = -2\sqrt{3}\n]", "This problem beautifully illustrates how logarithmic equations transform into exponential form, and emphasizes the importance of checking domain restrictions. Always verify your answers against the original logarithmic expression’s domain to ensure mathematical integrity.", "---", "### Key Takeaways:", "- Use the definition ( \log_b(a) = c \Rightarrow a = b^c ).\n- Solve the resulting algebraic equation carefully.\n- Check domain restrictions: ( x^2 - 4 > 0 ).\n- Both positive and negative roots must be evaluated.", "Mastering this method unlocks deeper understanding of logarithmic relationships and strengthens algebraic problem-solving skills.", "---", "Keywords for SEO:\nlogarithmic equations, solve log base 2 equation, log base 2 x² – 4 = 3, step-by-step logarithmic solution, x² – 4 = 8, determine domain of log, log equation solutions, algebra tip, exponential form conversion."]