2^{n-1} > \frac{500}{3} \approx 166.67

["Understanding When 2^{n−1} Exceeds 500⁄3 ≈ 166.67: A Complete Guide", "When solving mathematical inequalities involving exponential expressions, clarity and precision are essential. One such problem many encounter is determining the smallest integer value of ( n ) for which:", "[\n2^{n-1} > \frac{500}{3} \approx 166.67\n]", "This article explains how to solve this inequality step-by-step, highlights its real-world relevance, and explains key mathematical concepts to help you master exponential growth and inequality solving.", "---", "### What Is the Inequality We’re Solving?", "We want to find the smallest integer ( n ) such that:", "[\n2^{n-1} > 166.67 \quad (\ ext{since } \frac{500}{3} \approx 166.67)\n]", "This inequality compares exponential growth—specifically powers of 2—with a fixed decimal threshold.", "---", "### Step 1: Convert to a More Manageable Form", "To solve ( 2^{n-1} > 166.67 ), take the base-2 logarithm of both sides:", "[\nn - 1 > \log_2(166.67)\n]", "Since most calculators don’t directly compute ( \log_2 ), use the change-of-base formula:", "[\n\log_2(166.67) = \frac{\log_{10}(166.67)}{\log_{10}(2)} \approx \frac{2.222}{0.3010} \approx 7.384\n]", "So,", "[\nn - 1 > 7.384 \quad \Rightarrow \quad n > 8.384\n]", "---", "### Step 2: Find the Smallest Integer ( n )", "Since ( n ) must be an integer, the smallest value satisfying ( n > 8.384 ) is:", "[\nn = 9\n]", "---", "### Verification: Plug ( n = 9 ) Back into the Original Inequality", "[\n2^{9-1} = 2^8 = 256\n]", "Since ( 256 > 166.67 ), the inequality holds for ( n = 9 ).", "Check ( n = 8 ) for completeness:", "[\n2^{8-1} = 2^7 = 128 \quad \ ext{and} \quad 128 < 166.67\n]", "So ( n = 8 ) does not satisfy the inequality.", "---", "### Why This Inequality Matters: Growth and Thresholds", "Exponential expressions like ( 2^{n-1} ) model rapid growth—seen in computing (binary systems, algorithm complexity), population dynamics, and compound interest. Understanding when such expressions surpass a threshold helps in:", "- Algorithm analysis: Estimating computation time based on input size.\n- Finance: Calculating investment growth paths.\n- Bioinformatics: Tracking cell division or viral spread.", "In this case, knowing ( n = 9 ) means that by the 9th step, a doubling process exceeds a key benchmark of 166.67 units.", "---", "### Final Summary", "For the inequality:", "[\n2^{n-1} > \frac{500}{3} \approx 166.67\n]", "The smallest integer ( n ) satisfying the inequality is ( n = 9 ).", "---", "### Pro Tips for Solving Exponential Inequalities", "- Use logarithms to bring exponents down.\n- Always verify your answer by plugging it back.\n- Recognize common powers of 2 to speed up mental calculations.\n- Understand the context—exponential growth accelerates quickly!", "---", "### Key Takeaways", "- ( 2^{n-1} > 166.67 ) → find smallest ( n )\n- ( \log_2(166.67) \approx 7.384 ) → so ( n - 1 > 7.384 ) → ( n > 8.384 )\n- Smallest integer: ( \boxed{9} )", "---", "Understanding exponential inequalities equips you to predict growth thresholds across science, tech, and finance. Master this concept—it opens doors to solving real-world complex growth problems."]