The exponential function, specifically with base e (Euler's number, approximately 2.71828), is one of the most fundamental functions in mathematics and has widespread applications across various fields.
What is e^x?
The expression ex represents a number raised to the power of x, where the base is Euler's number, e. This function has a unique property: its rate of growth is proportional to its current value.
When x = 0, e0 = 1.
As x increases, ex grows rapidly.
As x decreases (becomes more negative), ex approaches 0.
Mathematical Definition
Mathematically, ex can be defined in several ways, including the limit:
ex = lim (1 + x/n)n as n approaches infinity.
It can also be represented by its Taylor series expansion around 0:
ex = Σ (xn / n!) for n from 0 to infinity (i.e., 1 + x/1! + x2/2! + x3/3! + ...)
Applications of e^x
The exponential function is crucial in modeling various natural phenomena and financial concepts:
Compound Interest: Continuously compounded interest is modeled using ert, where r is the interest rate and t is time.
Population Growth: In ideal conditions, populations (bacteria, animals, etc.) grow exponentially. The model is often P(t) = P0ekt, where P0 is the initial population, k is the growth rate constant, and t is time.
Radioactive Decay: The amount of a radioactive substance remaining after time t follows an exponential decay model: N(t) = N0e-λt, where N0 is the initial amount and λ is the decay constant.
Physics: Concepts like charging/discharging capacitors, cooling/heating processes, and damping oscillations often involve exponential functions.
Probability and Statistics: The normal distribution (bell curve) is closely related to the exponential function.
How This Calculator Works
This calculator takes a single input value, x, and computes the value of ex using the built-in JavaScript Math.exp() function. This function efficiently calculates the exponential value for the provided exponent.
Example: If you enter 2 for the Exponent (x), the calculator will compute e2, which is approximately 7.389.
function calculateExponent() {
var exponentInput = document.getElementById("exponentValue");
var resultDisplay = document.getElementById("result");
var exponentValue = parseFloat(exponentInput.value);
if (isNaN(exponentValue)) {
resultDisplay.innerHTML = "Please enter a valid number for the exponent.";
return;
}
var result = Math.exp(exponentValue);
// Format the result to a reasonable number of decimal places for readability
// You can adjust the number of decimal places as needed.
var formattedResult = result.toFixed(6); // For example, showing up to 6 decimal places
resultDisplay.innerHTML = "e" + exponentValue + " = " + formattedResult + "";
}