Solve the logarithmic equation \(\log_2(x+3) + \log_2(x-1) = 3\).
["# Solve the Logarithmic Equation: (\log_2(x+3) + \log_2(x-1) = 3)", "Solving logarithmic equations can seem challenging at first, but with the right steps, you can simplify and solve them efficiently. In this article, we’ll carefully solve the equation:", "[\n\log_2(x+3) + \log_2(x-1) = 3\n]", "## Understanding the Equation", "We begin by recognizing that this equation contains the sum of two logarithms with base 2. Using the logarithmic property:", "[\n\log_b A + \log_b B = \log_b(AB)\n]", "we combine the left-hand side:", "[\n\log_2[(x+3)(x-1)] = 3\n]", "This simplified form is easier to work with.", "## Rewriting in Exponential Form", "Recall that if (\log_b(M) = c), then (M = b^c). Applying this:", "[\n(x+3)(x-1) = 2^3\n]", "Since (2^3 = 8), we get:", "[\n(x+3)(x-1) = 8\n]", "## Expand and Simplify", "Expand the left-hand side:", "[\nx^2 - x + 3x - 3 = 8\n]\n[\nx^2 + 2x - 3 = 8\n]", "Subtract 8 from both sides:", "[\nx^2 + 2x - 11 = 0\n]", "## Solve the Quadratic Equation", "We now solve the quadratic:", "[\nx^2 + 2x - 11 = 0\n]", "Using the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "where (a = 1), (b = 2), (c = -11):", "[\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-11)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 44}}{2} = \frac{-2 \pm \sqrt{48}}{2}\n]", "Simplify (\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}):", "[\nx = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3}\n]", "So the two potential solutions are:", "[\nx = -1 + 2\sqrt{3} \quad \ ext{and} \quad x = -1 - 2\sqrt{3}\n]", "## Check for Validity in the Original Equation", "Logarithmic functions are only defined for positive arguments. So we must ensure:", "- (x + 3 > 0 \Rightarrow x > -3)\n- (x - 1 > 0 \Rightarrow x > 1)", "Now evaluate both solutions:", "1. (x = -1 + 2\sqrt{3} \approx -1 + 3.464 = 2.464 > 1), so valid\n2. (x = -1 - 2\sqrt{3} \approx -1 - 3.464 = -4.464 < 1), invalid", "Therefore, only (x = -1 + 2\sqrt{3}) is valid.", "## Final Answer", "The solution to the equation (\log_2(x+3) + \log_2(x-1) = 3) is:", "[\n\boxed{x = -1 + 2\sqrt{3}}\n]", "This value satisfies the original equation and lies within the domain of the logarithmic expressions.", "---", "### Key Takeaways", "- Use logarithmic identities to combine expressions.\n- Convert logarithmic equations to their exponential form.\n- Always verify solutions by checking the domain of the original equation.\n- Solving quadratics is a common final step in logarithmic problems.", "Mastering this process helps tackle more complex logarithmic equations with confidence."]
