Using logarithmic identity: \( \log_2[(x - 1)(x + 1)] = 3 \Rightarrow \log_2(x^2 - 1) = 3 \).

["Using Logarithmic Identity: ( \log_2[(x - 1)(x + 1)] = 3 \Rightarrow \log_2(x^2 - 1) = 3 )", "Understanding logarithmic expressions and identities is essential for solving equations involving logarithms efficiently. One crucial identity frequently applied in algebra and calculus is:", "[\n\log_b[(x - 1)(x + 1)] = y \quad \ ext{implies} \quad \log_b(x^2 - 1) = y\n]", "This identity stems from the logarithmic property that the logarithm of a product equals the sum of the logarithms:\n[\n\log_b[(x - 1)(x + 1)] = \log_b(x - 1) + \log_b(x + 1)\n]\nHowever, the more useful property for our purpose is:\n[\n\log_b[(x - 1)(x + 1)] = \log_b(x^2 - 1)\n]\nbecause ( (x - 1)(x + 1) = x^2 - 1 ).", "### Applying the Identity to the Given Equation", "Start with the given logarithmic equation:\n[\n\log_2[(x - 1)(x + 1)] = 3\n]", "Using the logarithmic identity:\n[\n\log_2(x^2 - 1) = 3\n]", "This transformation simplifies the expression, making it easier to solve for ( x ). The equation now focuses only on the argument of the logarithm: ( x^2 - 1 ), rather than its factored form—a key advantage when isolating ( x ).", "### Solving the Transformed Equation", "Given:\n[\n\log_2(x^2 - 1) = 3\n]", "To eliminate the logarithm, rewrite the equation in exponential form:\n[\nx^2 - 1 = 2^3 = 8\n]", "Solve for ( x^2 ):\n[\nx^2 = 8 + 1 = 9\n]", "Thus:\n[\nx = \pm 3\n]", "### Verifying the Domain Restrictions", "Before accepting the solutions, verify the domain of the original logarithmic expression. Since the argument ( x^2 - 1 ) must be positive (logarithm is undefined for non-positive arguments), requiring:\n[\nx^2 - 1 > 0 \Rightarrow x^2 > 1\n]", "Both ( x = 3 ) and ( x = -3 ) satisfy ( x^2 = 9 > 1 ), so both are valid solutions.", "### Practical Importance of the Identity", "This logarithmic identity is valuable in:", "- Simplifying expressions: Converting products inside logs to a single argument (( x^2 - 1 )) streamlines solving and manipulation.\n- Problem-solving efficiency: Avoiding expansion or breakdown of products allows quicker application of logarithmic rules and algebraic simplifications.\n- Understanding logarithmic structure: Reinforces how algebraic identities preserve meaningful meaning within logarithmic calculations.", "### Summary", "The identity connecting factored and product forms via logarithms—\n[\n\log_b[(x - 1)(x + 1)] = \log_b(x^2 - 1)\n]\n—enables clearer handling of logarithmic equations. Applying it to\n[\n\log_2[(x - 1)(x + 1)] = 3\n]\nsimplifies directly to ( \log_2(x^2 - 1) = 3 ), facilitating easy solving and broadening insight into logarithmic transformations.", "---", "Key Takeaways:\n- Recognize ( (x - 1)(x + 1) = x^2 - 1 ) to convert product logs to single argument logs.\n- This identity maintains equality while simplifying equation solving.\n- Always check that the argument remains positive to ensure domain consistency.\n- Use this method to wipe complexity and solve logarithmic equations faster and more accurately."]