Solve \( 25000(0.85)^t < 10000 \)
["Solve ( 25000(0.85)^t < 10000 ): A Step-by-Step Mathematical Guide", "When solving exponential inequalities like ( 25000(0.85)^t < 10000 ), it's essential to understand how exponential decay affects the inequality. This article walks through solving the inequality step-by-step, providing clarity and practical insight for students, math learners, and anyone interested in working with exponential expressions.", "---", "### Understanding the Inequality", "We are asked to solve:", "[\n25000(0.85)^t < 10000\n]", "This models a real-world scenario such as population decline, radioactive decay, or financial depreciation—anything that decreases by a constant percentage each time unit ( t ).", "---", "### Step 1: Isolate the Exponential Term", "Begin by dividing both sides of the inequality by 25,000 to simplify:", "[\n(0.85)^t < \frac{10000}{25000}\n]", "[\n(0.85)^t < 0.4\n]", "Now the inequality becomes:", "[\n(0.85)^t < 0.4\n]", "---", "### Step 2: Apply Logarithms", "Since the variable ( t ) appears in the exponent, the most reliable method is to apply logarithms. Recall the key property:", "[\n\log(a^t) = t \log a\n]", "Apply the logarithm (base 10 or natural log both work—results are equivalent):", "[\n\log\left( (0.85)^t \right) < \log(0.4)\n]", "[\nt \log(0.85) < \log(0.4)\n]", "Because ( 0.85 < 1 ), its logarithm is negative:\n( \log(0.85) < 0 )", "When dividing both sides of an inequality by a negative number, the direction of the inequality reverses. So divide both sides by ( \log(0.85) ), remembering to reverse the inequality:", "[\nt > \frac{\log(0.4)}{\log(0.85)}\n]", "---", "### Step 3: Calculate the Numerical Value", "Using a calculator:", "- ( \log(0.4) \approx -0.39794 )\n- ( \log(0.85) \approx -0.07058 )", "[\nt > \frac{-0.39794}{-0.07058} \approx 5.646\n]", "---", "### Final Answer", "[\nt > 5.646\n]", "Thus, the inequality holds for all ( t ) greater than approximately 5.646. Since ( t ) typically represents time in whole units, we interpret this as:", "The inequality is true for ( t \geq 6 ) if ( t ) is measured in discrete steps.", "---", "### Summary in Mathematical Form", "[\n\boxed{ t > \frac{\log(0.4)}{\log(0.85)} \approx 5.646 }\n]", "---", "### Applications and Interpretation", "This result tells us that after approximately 5.646 time units, the original quantity (25,000 times decay factor ( 0.85^t )) drops below 10,000. Understanding such thresholds is crucial in fields like finance, biology, engineering, and technology.", "---", "### Further Tips for Solving Exponential Inequalities", "- Always isolate the exponential expression first.\n- Remember that dividing or multiplying both sides by a negative number reverses the inequality.\n- Use logarithms when ( t ) is in the exponent.\n- Use a scientific calculator for accurate logarithmic values.", "---", "Key takeaway: Mastering exponential inequalities empowers you to model decay processes and solve real-world problems confidently. With practice, solving inequalities like ( 25000(0.85)^t < 10000 ) becomes straightforward and intuitive.", "---", "Keywords for SEO: solve ( 25000(0.85)^t < 10000 ), exponential inequality, logarithmic steps, mathematical solution, decay model, solve exponential inequality, logarithm tips, real-world application.", "---", "By applying logarithmic transformation carefully and interpreting results in context, you can confidently solve exponential inequalities like this one."]
