\[ 200 = 100 e^{0.05t} \]

["Understanding and Solving the Equation: 200 = 100 e^{0.05t}", "Mathematics frequently appears in real-world problems involving growth, decay, and exponential trends. One commonly encountered equation is:", "[\n200 = 100 e^{0.05t}\n]", "This formula describes exponential growth and is widely used in finance, biology, physics, and many other fields. In this article, we’ll walk through how to solve this equation step-by-step, interpret its meaning, and explore practical applications.", "---", "### What Does the Equation Represent?", "The equation\n[\n200 = 100 e^{0.05t}\n]\nmodels a situation where a quantity grows exponentially over time ( t ), starting at 100 and increasing at a continuous rate of 5% per unit time, resulting in 200 after time ( t ).", "---", "### Step-by-Step Solution", "We solve for ( t ):", "1. Isolate the exponential term:\nDivide both sides by 100:\n[\n\frac{200}{100} = e^{0.05t}\n]\n[\n2 = e^{0.05t}\n]", "2. Take the natural logarithm (ln) of both sides:\nThis eliminates the exponential base:\n[\n\ln(2) = \ln\left(e^{0.05t}\right)\n]", "3. Use the logarithmic identity ( \ln(e^x) = x ):\n[\n\ln(2) = 0.05t\n]", "4. Solve for ( t ):\n[\nt = \frac{\ln(2)}{0.05}\n]", "Since ( \ln(2) \approx 0.6931 ),\n[\nt \approx \frac{0.6931}{0.05} = 13.862\n]", "So,\n[\nt \approx 13.86 \ ext{ (units depending on context)}\n]", "---", "### Interpretation and Practical Use", "Solving ( 200 = 100 e^{0.05t} ) means finding the time at which an initially valued 100 grows to 200 with a continuous growth rate of 5% per unit time. This concept applies in:", "- Financial investments: Calculating how long a sum grows under continuous compound interest at 5% annual rate.\n- Population dynamics: Estimating how long it takes for a population to double given a 5% growth rate.\n- Radioactive decay or chemical reactions: When modeling exponential rise or decay.", "---", "### Visualizing the Exponential Growth", "Plotting ( y = 100 e^{0.05t} ), the curve starts at 100 and rises smoothly beyond 200 at ( t \approx 13.86 ), confirming our solution.", "---", "### Final Thoughts", "Understanding how to solve equations of the form ( A = B e^{rt} ) is crucial for anyone working with growth models. The ability to isolate ( t ) enables precise predictions in science and finance. Whether you're forecasting investment returns or modeling biological populations, equations like this provide the foundation for informed decision-making.", "---", "Keywords: \nExponentialGrowth #SolveExponentialEquation #200Equals100e0.05t #ExponentialModel #MathematicalSolution #ContinuousGrowth #NaturalLogarithm #tValueCalculation #MathExplanation", "Meta Description:\nLearn how to solve ( 200 = 100 e^{0.05t} ) step-by-step. Discover the time ( t ) required for growth with a 5% continuous rate and apply this to finance, biology, and science."]