Combine logs: \( \log_2(x(x - 3)) = 2 \).

["# Solving Combine Logs: ( \log_2(x(x - 3)) = 2 ) – A Step-by-Step Guide", "Working with logarithmic equations can feel challenging at first, but with the right approach, combining logs is straightforward. In this article, we’ll explore how to solve the equation:", "[\n\log_2(x(x - 3)) = 2\n]", "We’ll break down the process into simple, clear steps, explain the logic behind each transformation, and highlight key mathematical principles. Whether you're preparing for exams, tackling homework, or deepening your understanding, this guide will help you confidently solve combine logs.", "---", "## What Does the Equation Mean?", "The equation ( \log_2(x(x - 3)) = 2 ) states that the base-2 logarithm of the expression ( x(x - 3) ) equals 2. By the definition of logarithms, this means:", "[\nx(x - 3) = 2^2\n]", "which simplifies to:", "[\nx(x - 3) = 4\n]", "This transformation is the key step in solving combine logs — recognizing that ( \log_b(A) = C ) implies ( A = b^C ).", "---", "## Step 1: Expand and Rearrange the Equation", "Begin by expanding the product on the left-hand side:", "[\nx^2 - 3x = 4\n]", "Next, move all terms to one side to form a standard quadratic equation:", "[\nx^2 - 3x - 4 = 0\n]", "---", "## Step 2: Solve the Quadratic Using Factoring", "Now, factor the quadratic expression. We seek two numbers whose product is ( -4 ) and whose sum is ( -3 ). These numbers are ( -4 ) and ( +1 ):", "[\n(x - 4)(x + 1) = 0\n]", "---", "## Step 3: Find the Roots", "Set each factor equal to zero:", "[\nx - 4 = 0 \quad \Rightarrow \quad x = 4\n]\n[\nx + 1 = 0 \quad \Rightarrow \quad x = -1\n]", "So, the potential solutions are ( x = 4 ) and ( x = -1 ).", "---", "## Step 4: Verify Solutions in the Original Equation", "Because logarithms are only defined for positive arguments, we must check which solutions are valid.", "- For ( x = 4 ):\n ( x(x - 3) = 4 \cdot (4 - 3) = 4 ), which is positive → valid.\n Also, ( \log_2(4) = 2 ), which matches the right-hand side.", "- For ( x = -1 ):\n ( x(x - 3) = (-1)(-1 - 3) = (-1)(-4) = 4 ), also positive → valid numerically.\n However, check the logarithmic argument: ( x(x-3) = 4 > 0 ), so this value is allowed by domain.", "But important: Although both satisfy the algebraic form, we must ensure ( x(x - 3) > 0 ). Since both ( x = 4 ) and ( x = -1 ) yield positive values, both are mathematically valid. However, in real-world applications (especially in computing or measurements), domain restrictions may apply depending on context.", "---", "## Final Solution", "[\n\boxed{x = 4 \quad \ ext{or} \quad x = -1}\n]", "Both values satisfy the original equation.", "---", "## Why Understanding Combine Logs Matters", "Solving equations involving ( \log_b(A \cdot B) ) or similar combined logs helps develop algebraic fluency. Recognizing identities like:", "[\n\log_b(A \cdot C) = \log_b(A) + \log_b(C)\n]", "(later used when breaking down logs) and maintaining domain validity are essential skills in advanced math, science, and programming.", "---", "## Summary", "- Convert the logarithmic equation using exponentiation: ( x(x - 3) = 2^2 ).\n- Rearrange into standard quadratic form ( x^2 - 3x - 4 = 0 ).\n- Factor to find roots: ( x = 4 ), ( x = -1 ).\n- Verify domain: both arguments ( x(x-3) ) are positive.\n- Final solutions: both values are valid.", "---", "## Further Reading", "- Learn how to solve logarithmic equations with exponents.\n- Explore logarithmic identities and their proof.\n- Practice combining logs with product and quotient rules: ( \log_b(A \cdot C) = \log_b(A) + \log_b(C) ).\n- Discover method issues with complex numbers and invalid domains.", "---", "Mastering combine logs opens doors to precise modeling and scientific computation — keep practicing, and soon these steps will feel second nature!"]