We want to find \(n\) such that: \(50 \times (1.02)^n \geq 90\)

["How to Solve ( 50 \ imes (1.02)^n \geq 90 ) – Step-by-Step Guide", "Finding the smallest integer ( n ) such that ( 50 \ imes (1.02)^n \geq 90 ) requires understanding exponential equations and logarithms. This problem commonly appears in finance, population growth models, and compound interest calculations. In this article, we’ll break down how to solve this inequality step by step and explain the math behind it for clear, intuitive understanding.", "---", "### Understanding the Problem", "You are given the inequality:", "[\n50 \ imes (1.02)^n \geq 90\n]", "Your goal is to find the smallest integer ( n ) such that this inequality holds true. This models a situation where an initial value of 50 grows at 2% per period (monthly, quarterly, etc.), and you want to determine after how many periods it reaches or exceeds 90.", "---", "### Step 1: Isolate the Exponential Term", "Divide both sides of the inequality by 50 to isolate the exponential expression:", "[\n(1.02)^n \geq \frac{90}{50}\n]", "[\n(1.02)^n \geq 1.8\n]", "Now you need to solve for ( n ) in the equation ( (1.02)^n \geq 1.8 ).", "---", "### Step 2: Apply Logarithms to Solve for ( n )", "Since the variable ( n ) appears in the exponent, take the natural logarithm (ln) of both sides:", "[\n\ln\left((1.02)^n\right) \geq \ln(1.8)\n]", "Use the logarithmic identity ( \ln(a^b) = b \ln(a) ):", "[\nn \cdot \ln(1.02) \geq \ln(1.8)\n]", "Now divide both sides by ( \ln(1.02) ). Since ( \ln(1.02) > 0 ), the inequality direction remains unchanged:", "[\nn \geq \frac{\ln(1.8)}{\ln(1.02)}\n]", "---", "### Step 3: Compute the Numerical Value", "Calculate both logarithms:", "- ( \ln(1.8) \approx 0.5878 )\n- ( \ln(1.02) \approx 0.0198 )", "So,", "[\nn \geq \frac{0.5878}{0.0198} \approx 29.68\n]", "---", "### Step 4: Determine the Smallest Integer ( n )", "Since ( n ) must be an integer (number of periods), round up to the nearest whole number:", "[\nn = 30\n]", "---", "### Final Answer", "The smallest integer ( n ) satisfying\n[\n50 \ imes (1.02)^n \geq 90\n]\nis ( \mathbf{n = 30} ).", "---", "### Real-World Applications", "This type of calculation is essential in:\n- Financial planning: determining how long it takes for an investment to grow given interest rates.\n- Population studies: projecting growth over time at a constant rate.\n- Business forecasting: predicting sales or revenue growth under steady expansion.", "---", "### Conclusion", "Solving exponential inequalities combines algebraic manipulation with logarithms. By isolating the exponential term and applying logarithms, you efficiently determine the minimum time ( n ) required for a quantity growing at 2% per period to reach 90 from 50. This approach provides clear, actionable insight for both students and professionals working with exponential growth.", "---", "Keywords: solve ( 50 \ imes (1.02)^n \geq 90 ), exponential inequality, logarithms, growth calculation, compound interest, find integer ( n )", "---", "For further reading, explore how continuous compounding affects such inequalities, or how to graph exponential functions to visualize threshold crossings."]