Combine logs: \( \log_2((x+1)(x-1)) = 3 \).

["SEO-Optimized Article: Solve ( \log_2((x+1)(x-1)) = 3 ) – Step-by-Step Guide", "# Solve ( \log_2((x+1)(x-1)) = 3 ): The Complete Step-by-Step Solution", "If you're trying to solve logarithmic equations like ( \log_2((x+1)(x-1)) = 3 ), you’re in the right place. This article will guide you through how to solve this logarithmic equation, explain the key math involved, and provide helpful context for students, math learners, and problem solvers seeking clarity. Plus, we’ll optimize this post for search engines with targeted keywords and structured content.", "---", "## Understanding the Equation: ( \log_2((x+1)(x-1)) = 3 )", "### What it means\nThis equation involves a logarithm with base 2 of a product: ( (x+1)(x-1) ), set equal to 3. Our goal is to find all real values of ( x ) that make this true.", "---", "## Step 1: Simplify the Expression Inside the Log", "The product ( (x+1)(x-1) ) is a classic algebraic identity:\n[\n(x+1)(x-1) = x^2 - 1\n]\nSo, the equation becomes:\n[\n\log_2(x^2 - 1) = 3\n]", "Why this matters:\nLogarithmic equations are undefined when the argument is non-positive, so we must ensure ( x^2 - 1 > 0 ), later validating our solution.", "---", "## Step 2: Eliminate the Logarithm by Exponentiation", "Recall that:\nIf ( \log_b(A) = C ), then ( A = b^C ).", "Apply this rule:\n[\nx^2 - 1 = 2^3 = 8\n]", "---", "## Step 3: Solve the Resulting Quadratic Equation", "From:\n[\nx^2 - 1 = 8 \implies x^2 = 9 \implies x = \pm 3\n]", "---", "## Step 4: Check for Validity in the Original Equation", "Substitute ( x = 3 ) and ( x = -3 ) into the original expression:", "- For ( x = 3 ):\n[\n(x+1)(x-1) = (4)(2) = 8 \Rightarrow \log_2(8) = 3 \quad \ ext{(Valid)}\n]", "- For ( x = -3 ):\n[\n(x+1)(x-1) = (-2)(-4) = 8 \Rightarrow \log_2(8) = 3 \quad \ ext{(Valid)}\n]", "Both values satisfy the original equation.", "Important note:\nSince ( x = \pm 3 ) both make the argument positive (( 8 > 0 )), both are acceptable solutions — no extraneous roots here.", "---", "## Practical Tips: When Solving Log Equations", "- Always convert the logarithmic equation to its exponential form.\n- Check domain restrictions: the argument of log must be positive.\n- Simplify expressions first (e.g., using identities).\n- Always verify solutions in the original equation.\n- Be cautious with negative values — they may cancel out but don’t invalidate log as long as the result is positive.", "---", "## Why Solve ( \log_2((x+1)(x-1)) = 3 )?", "This type of logarithmic equation appears in calculus (logarithmic differentiation), physics (audibility levels in dB), and computer science (algorithmic complexity). Mastering these steps builds your foundation for advanced math and real-world applications.", "---", "## Key SEO Keywords for This Article", "- ( \log_2((x+1)(x-1)) = 3 )\n- Solve logarithmic equations step-by-step\n- How to solve ( \log_b(x^2 - 1) = 3 )\n- Combine logs: simplifying ( (x+1)(x-1) )\n- Domain of logarithmic function ( \log_2(x^2 - 1) )\n- Solve ( \log_2(x^2 - 1) = 3 \ with verification\n- Step-by-step logarithmic equations guide", "---", "## Summary: Final Answer", "The solutions to ( \log_2((x+1)(x-1)) = 3 ) are:\n[\n\boxed{x = -3 \quad} \ ext{and} \quad x = 3\n]", "Both are valid, real, and satisfy the original logarithmic equation.", "---", "## Frequently Asked Questions (FAQ)", "Q: Can ( x ) be negative in this equation?\nA: Yes, ( x = -3 ) works because ( (x+1)(x-1) > 0 ) for both ( x = -3 ) and ( x = 3 ), keeping the logarithm defined.", "Q: What if the argument was negative?\nA: The logarithm is undefined for negative numbers, so no real solutions exist in that case.", "Q: How do I know if ( x^2 - 1 > 0 )?\nA: Solve the inequality ( x^2 - 1 > 0 ) → ( x < -1 ) or ( x > 1 ). Both ( x = -3 ) and ( x = 3 ) satisfy this.", "---", "### Start solving logarithmic equations today — master the basics, verify carefully, and expand your math skills!", "---", "Meta Tags for SEO Optimization:", "html\n<title>Solve \( \log_2((x+1)(x-1)) = 3 \) – Step-by-Step Algebra Guide</title>\n<meta name="description" content="Learn how to solve \( \log_2((x+1)(x-1)) = 3 \) with clear steps, domain checks, and verification. Perfect for students and math enthusiasts.">\n<meta name="keywords" content="logarithmic equations, solve log base 2, \( \log_2((x+1)(x-1)) = 3 \), step-by-step math, quadratic solution, domain of log">", "---", "If you found this guide helpful, share it and explore more math tutorials optimized for search engines!"]