Solve for \( x \) in the equation \( \log_2(x+1) + \log_2(x-1) = 3 \).
["# Solve for ( x ) in the Equation ( \log_2(x+1) + \log_2(x-1) = 3 ): A Step-by-Step Guide", "Understanding logarithmic equations can sometimes feel challenging, but solving equations like ( \log_2(x+1) + \log_2(x-1) = 3 ) becomes easier when broken down step-by-step. In this article, we will solve the equation ( \log_2(x+1) + \log_2(x-1) = 3 ) and uncover the correct value(s) of ( x ) that satisfy it.", "## Understanding the Equation", "The equation is:\n[\n\log_2(x+1) + \log_2(x-1) = 3\n]\nThis involves the sum of two logarithmic terms with base 2. Our goal is to isolate ( x ) by applying logarithmic properties and algebraic manipulation.", "### Step 1: Use the Logarithmic Property for Sum", "Recall the logarithmic identity:\n[\n\log_b A + \log_b B = \log_b (A \cdot B)\n]\nApplying this to our equation gives:\n[\n\log_2\left( (x+1)(x-1) \right) = 3\n]\nNote: The logarithm is only defined for positive arguments, so ( x+1 > 0 ) and ( x-1 > 0 ). These conditions guide our domain restriction.", "### Step 2: Simplify the Argument", "Simplify ( (x+1)(x-1) ) using the difference of squares:\n[\n(x+1)(x-1) = x^2 - 1\n]\nSo the equation becomes:\n[\n\log_2(x^2 - 1) = 3\n]", "### Step 3: Eliminate the Logarithm", "To remove the logarithm, rewrite the equation in exponential form:\n[\nx^2 - 1 = 2^3\n]\n[\nx^2 - 1 = 8\n]\n[\nx^2 = 9\n]\n[\nx = \pm 3\n]", "### Step 4: Check for Valid Solutions", "Recall from Step 1 that ( x+1 > 0 ) and ( x-1 > 0 ), meaning:\n[\nx > 1\n]\nAmong ( x = 3 ) and ( x = -3 ), only ( x = 3 ) satisfies ( x > 1 ). The solution ( x = -3 ) leads to taking ( \log_2(-2) ), which is undefined.", "### Final Answer", "The only valid solution is:\n[\n\boxed{3}\n]", "---", "### Summary", "By combining logarithmic properties and carefully analyzing domain restrictions, we solve:\n[\n\log_2(x+1) + \log_2(x-1) = 3\n]\nleading uniquely to ( x = 3 ), verified through substitution and logical constraints. Mastering these steps improves problem-solving confidence with logarithmic equations.", "---", "Keywords: solve ( \log_2(x+1) + \log_2(x-1) = 3 ), logarithmic equation steps, how to solve log equations, step-by-step logarithm solution, ( x ) in log base 2, domain of logarithmic functions.", "Meta Description: Solve ( \log_2(x+1) + \log_2(x-1) = 3 ) step-by-step. Learn how to combine logs, eliminate logs, then check domain restrictions for valid solutions."]
