Question: A drug dosage model requires solving $ \log_2(x + 4) - \log_2(x - 1) = 3 $. Find the value of $ x $.

["Solving the Equation: A Drug Dosage Model Using Logarithmic Functions", "In pharmaceutical research and medicine, precise drug dosage models are essential to ensure patient safety and efficacy. One key step in modeling drug concentration over time involves solving logarithmic equations that describe how dosage levels behave under varying conditions. A common challenge in such models is solving logarithmic expressions involving base 2, particularly equations of the form:", "$$\n\log_2(x + 4) - \log_2(x - 1) = 3\n$$", "Understanding and solving this equation empowers developers and scientists to determine critical dosage thresholds accurately. In this article, we walk through the step-by-step solution of the equation, explaining how logarithmic properties simplify the expression and lead to the correct value of $ x $.", "---", "### Step 1: Combine Logarithms Using Properties", "Using the logarithm quotient rule, which states:", "$$\n\log_b A - \log_b B = \log_b\left(\frac{A}{B}\right)\n$$", "we can rewrite the original equation:", "$$\n\log_2\left(\frac{x + 4}{x - 1}\right) = 3\n$$", "This simplifies the expression significantly, converting subtraction inside logs into a division.", "---", "### Step 2: Eliminate the Logarithm", "To remove the base-2 logarithm, exponentiate both sides using base 2:", "$$\n2^{\log_2\left(\frac{x + 4}{x - 1}\right)} = 2^3\n$$", "Since $ 2^{\log_2(y)} = y $, this simplifies to:", "$$\n\frac{x + 4}{x - 1} = 8\n$$", "---", "### Step 3: Solve the Rational Equation", "Now solve:", "$$\n\frac{x + 4}{x - 1} = 8\n$$", "Multiply both sides by $ x - 1 $ (noting $ x <br/>\ne 1 $ to avoid undefined expressions):", "$$\nx + 4 = 8(x - 1)\n$$", "Expand the right-hand side:", "$$\nx + 4 = 8x - 8\n$$", "Bring all terms to one side:", "$$\n4 + 8 = 8x - x \quad \Rightarrow \quad 12 = 7x\n$$", "$$\nx = \frac{12}{7}\n$$", "---", "### Step 4: Validate the Solution", "Ensure the solution satisfies the original logarithmic expressions and domain conditions:", "- $ x + 4 = \frac{12}{7} + 4 = \frac{40}{7} > 0 $\n- $ x - 1 = \frac{12}{7} - 1 = \frac{5}{7} > 0 $", "Both arguments of the logarithms are positive, so the solution is valid within the domain.", "---", "### Conclusion: Practical Relevance in Drug Dosage Modeling", "Solving equations like $ \log_2(x + 4) - \log_2(x - 1) = 3 $ is not just a mathematical exercise—it’s crucial for developing reliable drug dosage models. These models help predict how drug concentrations stabilize or reach therapeutic levels in patients, guiding clinicians to prescribe accurate dosages.", "In real-world applications, such models may account for variable absorption, metabolism, and elimination rates—factors encoded mathematically through logarithmic relationships. Mastering these analytical techniques strengthens the foundation for innovation in pharmaceutical science and personalized medicine.", "Keywords: drug dosage model, logarithmic equation, solve $ \log_2(x + 4) - \log_2(x - 1) = 3 $, exponential and logarithmic functions in medicine, scientific problem solving."]