Solve: 100 × (0.95)^t < 50 → (0.95)^t < 0.5

["Solving the Exponential Inequality: 100 × (0.95)^t < 50 — Step-by-Step Guide", "Understanding exponential inequalities is essential in many fields, including finance, science, and engineering. One common problem is solving expressions of the form:", "> 100 × (0.95)ᵗ < 50", "This inequality asks: for what values of t is the quantity 100 multiplied by (0.95) raised to the power t less than 50?", "In this article, we’ll walk through how to solve this inequality step by step, explain the logic behind the solution, and provide clarity on interpreting exponential models.", "---", "### Step 1: Isolate the Exponential Term", "Start by isolating the exponential expression:", "[\n100 \ imes (0.95)^t < 50\n]", "Divide both sides by 100:", "[\n(0.95)^t < \frac{50}{100} = 0.5\n]", "Now the inequality becomes:", "> (0.95)^t < 0.5", "---", "### Step 2: Apply Logarithms to Both Sides", "Because the variable t appears in the exponent, the most effective method to solve for t is to use logarithms. Since the base of the exponential is 0.95, which is positive and less than 1, the function is decreasing. That criterion helps interpret the inequality correctly once solved.", "Take the natural logarithm (ln) of both sides:", "[\n\ln\left((0.95)^t\right) < \ln(0.5)\n]", "Using the logarithmic power rule, ( \ln(a^b) = b \ln(a) ), this becomes:", "[\nt \cdot \ln(0.95) < \ln(0.5)\n]", "---", "### Step 3: Solve for t and Reverse the Inequality", "Now, divide both sides by ( \ln(0.95) ). But note: ( \ln(0.95) ) is negative because 0.95 < 1. When dividing by a negative number, the inequality sign flips:", "[\nt > \frac{\ln(0.5)}{\ln(0.95)}\n]", "Calculate the numerical value:", "- ( \ln(0.5) \approx -0.6931 )\n- ( \ln(0.95) \approx -0.05129 )", "So:", "[\nt > \frac{-0.6931}{-0.05129} \approx 13.51\n]", "---", "### Final Answer", "The inequality ( 100 \ imes (0.95)^t < 50 ) holds when:", "[\nt > \frac{\ln(0.5)}{\ln(0.95)} \approx 13.51\n]", "---", "### Interpretation & Real-World Applications", "This result means that the quantity 100 times (0.95)^t drops below 50 only when t exceeds approximately 13.51. This has practical uses:", "- Finance: When modeling depreciation or asset value decline at a constant rate, this helps determine after how many periods a value falls below a target.\n- Science: Useful in radioactive decay, population decline, or drug concentration over time.\n- Engineering: Predicts when a system’s efficiency or signal strength drops below a threshold.", "Understanding exponential decay through inequalities empowers better decision-making in quantitative fields.", "---", "### Summary", "To solve 100 × (0.95)^t < 50:", "1. Isolate the exponential: ( (0.95)^t < 0.5 )\n2. Apply natural log to both sides\n3. Account for negative logarithms by flipping the inequality\n4. Solve for t: ( t > \frac{\ln(0.5)}{\ln(0.95)} \approx 13.51 )", "Mastering this technique sharpens your ability to analyze real-world decay processes governed by exponential functions.", "---", "Keywords: exponential inequality, solve 100 × (0.95)^t < 50, logarithmic inequality, exponential decay, t exponent solution, 0.95^t inequality, logarithms applied, real-world applications, mathematical modeling", "---", "Start applying these methods today — solving exponential inequalities opens doors to deeper insights in science, finance, and technology!"]