Find the value of \( x \) that satisfies \( \log_2(x^2 - 4) = 3 \).
["# Find the Value of ( x ) That Satisfies ( \log_2(x^2 - 4) = 3 )", "Understanding logarithmic equations is essential in algebra, and solving ( \log_2(x^2 - 4) = 3 ) is a straightforward yet insightful example. This article walks you through finding the value of ( x ) that satisfies the equation, explaining each step with clarity and mathematical rigor.", "## Understanding the Equation", "The equation given is:", "[\n\log_2(x^2 - 4) = 3\n]", "This logarithmic expression tells us that the logarithm base 2 of the expression ( x^2 - 4 ) equals 3. To solve for ( x ), we need to convert this logarithmic equation into its exponential form.", "## Step 1: Convert to Exponential Form", "Recall the fundamental identity connecting logarithms and exponents:", "[\n\log_b(a) = c \quad \Leftrightarrow \quad b^c = a\n]", "Applying this to our equation with base 2 exponent 3:", "[\nx^2 - 4 = 2^3\n]", "## Step 2: Simplify the Right-Hand Side", "Calculate ( 2^3 ):", "[\nx^2 - 4 = 8\n]", "## Step 3: Solve for ( x^2 )", "Add 4 to both sides:", "[\nx^2 = 8 + 4 = 12\n]", "## Step 4: Solve for ( x )", "Take the square root of both sides:", "[\nx = \pm\sqrt{12}\n]", "Simplify the square root:", "[\nx = \pm 2\sqrt{3}\n]", "## Step 5: Check Validity of Solutions", "Before finalizing, remember that logarithms are only defined for positive inputs. The argument of the logarithm is ( x^2 - 4 ), so we require:", "[\nx^2 - 4 > 0\n]", "Substitute ( x^2 = 12 ):", "[\n12 - 4 = 8 > 0 \quad \ ext{— valid}\n]", "Both ( x = 2\sqrt{3} ) and ( x = -2\sqrt{3} ) satisfy the domain condition, so both values are acceptable.", "## Final Answer", "The values of ( x ) that satisfy the equation are:", "[\nx = 2\sqrt{3} \quad \ ext{or} \quad x = -2\sqrt{3}\n]", "This equation exemplifies how to solve logarithmic equations by converting them to exponential form and carefully verifying the domain. Mastering these steps builds a strong foundation in logarithmic problem-solving.", "---", "Keywords:\nlog base 2 logarithm, solve log equation, find x such that log₂(x² − 4) = 3, algebraic equation solution, logarithmic domain, x² − 4 = 2³, √12 simplification, valid logarithmic values", "Meta description:\nLearn how to solve ( \log_2(x^2 - 4) = 3 ) step-by-step — including conversion to exponential form, solving for ( x ), and verifying the domain. Clear explanation with final answers."]
