Divide both sides by 50: \( 2^{t/3} \geq 16,000 \)

["Step-by-Step Solution: Divide Both Sides by 50 in ( 2^{t/3} \geq 16,000 )", "Understanding exponential inequalities is essential for solving many real-world problems in science, finance, and engineering. Today, we’ll walk through the process of solving the inequality ( 2^{t/3} \geq 16,000 ) by dividing both sides by 50 — and why this step is important in simplifying complex expressions.", "---", "### Why Divide Both Sides?", "Dividing both sides of an exponential inequality by a positive number preserves the inequality’s direction. Since 50 is positive, dividing simply helps rewrite the inequality in a more manageable form — in this case, reducing the constant on the right-hand side.", "---", "### Step 1: Divide Both Sides by 50", "Start with:\n[\n2^{t/3} \geq 16,000\n]", "Divide both sides by 50:\n[\n\frac{2^{t/3}}{50} \geq \frac{16,000}{50}\n]", "Calculate the division:\n[\n\frac{16,000}{50} = 320\n]", "So the inequality becomes:\n[\n2^{t/3} \geq 320\n]", "---", "### Step 2: Express 320 as a Power of 2 (or Use Logarithms)", "While 320 is not an exact power of 2, expressing or estimating powers helps refine the solution. Recall:\n[\n2^5 = 32,\quad 2^8 = 256,\quad 2^9 = 512\n]", "Since 320 lies between ( 2^8 = 256 ) and ( 2^9 = 512 ), we know:\n[\n2^8 < 320 < 2^9\n]", "Taking logarithms (base 2) of both sides:\n[\n\frac{t}{3} \geq \log_2(320)\n]", "Use change-of-base formula:\n[\n\log_2(320) = \frac{\log_{10}(320)}{\log_{10}(2)} \approx \frac{2.505}{0.3010} \approx 8.32\n]", "Multiply both sides by 3:\n[\nt \geq 3 \ imes 8.32 \approx 24.96\n]", "---", "### Final Solution", "Thus, the solution to the original inequality is:\n[\nt \geq 24.96 \quad (\ ext{approximately})\n]", "Or, more precisely:\n[\nt \geq 3 \log_2(320)\n]", "---", "### Practical Implications", "Dividing by 50 simplified the coefficient, enabling clearer logarithmic analysis. This method is valuable in modeling situations like population growth, radioactive decay, or investment doubling times—where exponential relationships dominate.", "---", "### Key Takeaways", "- Always verify the sign of divisors; positivity preserves inequality direction.\n- Dividing helps reduce constants, making logarithmic transformation feasible.\n- Combining algebraic manipulation with logarithmic tools unlocks exact and approximate solutions.", "---", "Keywords:\n( 2^{t/3} \geq 16,000 ), divide both sides by 50, exponential inequality, logarithms, solving inequalities with exponents, real-world applications.", "---", "Optimizing for SEO:\nThis article targets learners and professionals seeking clear, step-by-step guidance on solving exponential inequalities. Key phrases like “solve ( 2^{t/3} \geq 16,000 )”, “how to divide both sides of an exponential inequality,” and “logarithmic approach to exponential inequalities” are strategically embedded to boost search visibility."]