Exponential growth is a process where the quantity of a substance or population increases at a rate proportional to its current value. This means that the larger the quantity gets, the faster it grows. This type of growth is commonly observed in biological populations (like bacteria or animal herds under ideal conditions), financial investments, and the spread of certain diseases. The core idea is that the growth rate is not constant but rather a percentage of the current amount. The formula for exponential growth is often expressed as:
N(t) = N_0 * e^(rt)
Where:
N(t) is the quantity at time t.
N_0 is the initial quantity at time t=0.
e is the base of the natural logarithm (approximately 2.71828).
r is the exponential growth rate (per unit of time).
t is the time elapsed.
This calculator helps you determine the exponential growth rate r given an initial quantity, a final quantity, and the time elapsed. This rate is crucial for understanding how quickly a quantity is increasing and for making predictions about future values.
Calculate Exponential Growth Rate
function calculateGrowthRate() {
var initialQuantity = parseFloat(document.getElementById("initialQuantity").value);
var finalQuantity = parseFloat(document.getElementById("finalQuantity").value);
var timeElapsed = parseFloat(document.getElementById("timeElapsed").value);
var resultDiv = document.getElementById("result");
if (isNaN(initialQuantity) || isNaN(finalQuantity) || isNaN(timeElapsed)) {
resultDiv.innerHTML = "Please enter valid numbers for all fields.";
return;
}
if (initialQuantity <= 0) {
resultDiv.innerHTML = "Initial quantity must be greater than zero.";
return;
}
if (finalQuantity <= 0) {
resultDiv.innerHTML = "Final quantity must be greater than zero.";
return;
}
if (timeElapsed <= 0) {
resultDiv.innerHTML = "Time elapsed must be greater than zero.";
return;
}
// Formula derived from N(t) = N₀ * e^(rt)
// N(t) / N₀ = e^(rt)
// ln(N(t) / N₀) = rt
// r = ln(N(t) / N₀) / t
var ratio = finalQuantity / initialQuantity;
var naturalLogOfRatio = Math.log(ratio);
var growthRate = naturalLogOfRatio / timeElapsed;
resultDiv.innerHTML = "The exponential growth rate (r) is: " + growthRate.toFixed(4) + " per unit of time.";
}