Combine logs: log₂[(x + 3)(x − 1)] = 3 → (x + 3)(x − 1) = 2³ = 8.

["# Understanding Combine Logs: Solving log₂[(x + 3)(x − 1)] = 3 by Converting to Combinatorial Log Form", "When solving logarithmic equations, one of the most powerful techniques is converting logarithmic expressions into their equivalent exponential (combinatorial or power) forms. This approach not only simplifies complex equations but also enhances clarity, making it easier to solve for unknown variables. In this article, we’ll explore how to solve the logarithmic equation log₂[(x + 3)(x − 1)] = 3 by converting it to its combinatorial logarithmic form and systematically solving for ( x ).", "---", "## The Power of Logarithmic Conversion", "The logarithmic identity fundamental to solving such equations is:", "> logₐ(b) = c ⇔ aᶜ = b", "For any positive base ( a > 0 ), ( a <br/>\neq 1 ), this conversion allows us to eliminate the logarithm and work with a polynomial or simpler form. In this case, the base is 2, so:", "[\n\log₂[(x + 3)(x - 1)] = 3 \quad \ ext{is equivalent to} \quad (x + 3)(x - 1) = 2^3\n]", "Since ( 2^3 = 8 ), we directly rewrite the original equation as:", "[\n(x + 3)(x - 1) = 8\n]", "---", "## Step-by-Step Solution Using Combine Logs Logic", "### Step 1: Rewrite using logarithmic equivalence", "Start with the original:", "[\n\log₂[(x + 3)(x - 1)] = 3\n]", "Apply the logarithmic conversion rule:", "[\n(x + 3)(x - 1) = 2^3\n]", "### Step 2: Simplify the right-hand side", "[\n(x + 3)(x - 1) = 8\n]", "### Step 3: Expand the left-hand side", "[\nx \cdot x + x \cdot (-1) + 3 \cdot x + 3 \cdot (-1) = x^2 - x + 3x - 3 = x^2 + 2x - 3\n]", "So the equation becomes:", "[\nx^2 + 2x - 3 = 8\n]", "### Step 4: Bring all terms to one side to form a quadratic", "[\nx^2 + 2x - 3 - 8 = 0 \quad \Rightarrow \quad x^2 + 2x - 11 = 0\n]", "### Step 5: Solve the quadratic equation", "Apply the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad \ ext{where } a = 1, b = 2, c = -11\n]", "[\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 √48:", "[\n\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}\n]", "Thus:", "[\nx = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3}\n]", "---", "## Step 6: Verify domain restrictions", "Before accepting solutions, check the domain of the original logarithmic expression:", "[\n\log₂[(x + 3)(x - 1)] \quad \ ext{requires } (x + 3)(x - 1) > 0\n]", "Factor the expression:", "[\n(x + 3)(x - 1) > 0\n]", "The roots are ( x = -3 ) and ( x = 1 ). The parabola opens upward, so the expression is positive when:", "[\nx < -3 \quad \ ext{or} \quad x > 1\n]", "Now evaluate both solutions:", "- ( x = -1 + 2\sqrt{3} \approx -1 + 3.464 = 2.464 > 1 ) → valid\n- ( x = -1 - 2\sqrt{3} \approx -1 - 3.464 = -4.464 < -3 ) → valid", "Both solutions lie in the domain, so both are acceptable.", "---", "## Why This Method Works (Logic Behind Combine Logs)", "By converting logarithms to exponential form, we transform a multiplicative relationship inside the log into a straightforward equality. This leverages algebraic manipulation (expanding, simplifying, solving quadratics) to directly isolate the variable. The Key Takeaway: Logarithmic conversions preserve solution accuracy while simplifying computation.", "---", "## Real-World Application", "Equations of this form appear in science, engineering, and finance where growth or decay models rely on logarithmic scaling—such as in pH calculations, financial time value of money, or population models. Mastering logarithmic conversion empowers advanced problem-solving in these fields.", "---", "## Summary", "- Start with log₂[(x + 3)(x − 1)] = 3\n- Convert to exponential form: (x + 3)(x − 1) = 8\n- Expand and simplify to a quadratic: x² + 2x - 11 = 0\n- Use the quadratic formula to find: x = −1 ± 2√3\n- Verify both solutions lie in the domain: valid\n- Use logarithmic conversion logic to simplify, expand, and solve efficiently", "---", "Key search terms for SEO:\nlogarithmic equations, log base 2 conversion, solving log equations step-by-step, combine logs to simplify, rationalize quadratic from logs, domain restrictions logarithmic equations", "---", "Why invest time in Combine Logs strategy?\nIt simplifies complex logarithmic problems, improves algebra skills, and strengthens problem-solving strategies critical in STEM fields and competitive exams.", "---", "Discover more advanced logarithmic techniques and real-world applications at [Your Website or Blog Name], where every equation tells a story."]