Solve: 50 × (1.03)^t > 75 → (1.03)^t > 1.5
["Solve: 50 × (1.03)^t > 75 — How to Solve the Exponential Inequality (1.03)^t > 1.5", "In everyday life and advanced mathematics, solving exponential equations helps us model growth and decay — from savings interest to population increases. One common problem students and professionals face is solving exponential inequalities like:", "50 × (1.03)^t > 75", "But what does this really mean? How do we solve it? And why is it important?", "---", "### Step-by-Step Guide to Solving 50 × (1.03)^t > 75", "#### Step 1: Isolate the Exponential Term\nBegin by dividing both sides of the inequality by 50 to simplify:\n[\n(1.03)^t > \frac{75}{50} = 1.5\n]", "This turns the inequality into a standard exponential form:\n[\n(1.03)^t > 1.5\n]", "---", "#### Step 2: Use Logarithms to Solve for t\nSince the variable t appears in the exponent, we must use logarithms to bring t down:\nTake the natural logarithm (ln) of both sides:\n[\n\ln\left((1.03)^t\right) > \ln(1.5)\n]", "Using the logarithmic identity $\ln(a^b) = b \cdot \ln(a)$, we simplify:\n[\nt \cdot \ln(1.03) > \ln(1.5)\n]", "---", "#### Step 3: Solve for t and Account for the Sign of $\ln(1.03)$\nSince $\ln(1.03) > 0$, we can safely divide both sides by $\ln(1.03)$:\n[\nt > \frac{\ln(1.5)}{\ln(1.03)}\n]", "Now compute the values:\n- $\ln(1.5) \approx 0.4055$\n- $\ln(1.03) \approx 0.02956$\nSo:\n[\nt > \frac{0.4055}{0.02956} \approx 13.73\n]", "---", "### Final Answer:\n[\n\boxed{t > \log_{1.03}(1.5) \approx 13.73}\n]", "This means the inequality holds for all t greater than approximately 13.73 years, depending on the base interest rate.", "---", "### Why This Inequality Matters", "This type of exponential inequality commonly arises in finance when evaluating compound interest:\nIf you invest $50 at a 3% annual growth rate, after t years your amount becomes $50 × (1.03)^t. To reach at least $75, your investment must grow longer than about 13.73 years.", "Understanding how to solve such equations empowers informed financial decisions, scientific predictions, and economic planning.", "---", "### Quick Recap: Key Steps\n1. Isolate the exponential term: (1.03)^t > 1.5\n2. Apply logarithms to both sides\n3. Solve for t using logarithmic division\n4. Interpret the result in context", "---", "### Want More?\nExplore how to graph exponential functions, apply logarithms in real-world scenarios, or tackle complex inequalities — all critical tools in math, science, and finance.", "Keywords: exponential inequality, solve (1.03)^t > 1.5, log calculations, compound interest, t > log(1.5)/log(1.03), financial math solutions."]
