Solve: 20 × (0.95)^t < 5 → (0.95)^t < 0.25
["Solving the Inequality: 20 × (0.95)^t < 5\nHow to Solve Exponential Inequalities Step-by-Step", "When faced with exponential inequalities like ( 20 \ imes (0.95)^t < 5 ), understanding how to isolate the exponential term and solve for ( t ) is essential. In this article, we break down how to solve:", "[\n20 \ imes (0.95)^t < 5\n]", "Step 1: Simplify the Inequality\nBegin by dividing both sides by 20 to simplify:", "[\n(0.95)^t < \frac{5}{20}\n]", "[\n(0.95)^t < 0.25\n]", "Now, the inequality is in a standard form: an exponential expression ( 0.95^t ) is less than 0.25.", "Step 2: Apply Logarithms\nBecause ( t ) appears in the exponent, we use logarithms to bring ( t ) down. Since the base 0.95 is less than 1, the function is decreasing — so when taking logarithms, the inequality direction reverses:", "[\n\log(0.95^t) < \log(0.25)\n]", "Using the logarithmic identity ( \log(a^b) = b \log(a) ):", "[\nt \cdot \log(0.95) < \log(0.25)\n]", "Step 3: Solve for ( t )\nDivide both sides by ( \log(0.95) ). Note: since ( 0.95 < 1 ), ( \log(0.95) < 0 ), so dividing by a negative value reverses the inequality:", "[\nt > \frac{\log(0.25)}{\log(0.95)}\n]", "Using approximate values:", "[\n\log(0.25) \approx -0.6021, \quad \log(0.95) \approx -0.0223\n]", "[\nt > \frac{-0.6021}{-0.0223} \approx 27.03\n]", "Final Answer:\n[\nt > 27.03 \quad (\ ext{approximately})\n]", "That is, the inequality ( 20 \ imes (0.95)^t < 5 ) holds true for all ( t > 27.03 ). Most often rounded or expressed as:", "[\nt > \frac{\log(0.25)}{\log(0.95)} \approx 27.03\n]", "Why This Matters\nUnderstanding how to solve exponential inequalities helps in fields like finance (decay models), physics (radioactive decay), and computer science (algorithmic complexity). Mastering these steps allows for precise modeling and accurate predictions.", "---", "Summary: To solve ( 20 \ imes (0.95)^t < 5 ), divide by 20, isolate ( (0.95)^t < 0.25 ), then use logarithms (remembering the inequality reversal due to negative logs) to find ( t > \frac{\log(0.25)}{\log(0.95)} ). This technique is key for solving real-world exponential decay problems.", "Keywords: exponential inequality, solve (0.95)^t < 0.25, logarithmic inequality, mathematical problem solving, exponential decay, t in exponential, inequality steps, real number solution, t > 27.03", "---", "Advanced Tip: Use a graphing calculator or software to visualize ( y = 20 \ imes (0.95)^t ) and see where it crosses ( y = 5 )—angling your solution visually complements algebraic steps."]
