Enter a base and an exponent (x) to evaluate exponential functions of the form a^x or the natural exponential e^x. Choose your function type, supply your values, and get back the result, the natural log of the result, and a visual curve showing how the function grows.
f(x) Result
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Base Used
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ln(f(x))
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log₁₀(f(x))
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Function Expression
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An exponential function is a mathematical function of the form f(x) = a^x, where 'a' is a positive constant called the base and x is the exponent. The function grows (or decays) at a rate proportional to its current value. When the base is Euler's number e ≈ 2.71828, it's called the natural exponential function, f(x) = e^x.
The general exponential function formula is f(x) = a^x, where a > 0 and a ≠ 1. A common variant is f(x) = p · e^(kx), where p is the initial value and k is the growth or decay rate. When k > 0 the function grows; when k < 0 it decays.
The letter e represents Euler's number, approximately 2.718281828459. It is an irrational mathematical constant that serves as the base of the natural logarithm. The function e^x is unique because it is its own derivative, making it fundamental in calculus and natural growth models.
Given two points (x₁, y₁) and (x₂, y₂), you can find f(x) = a^x by solving: a = (y₂/y₁)^(1/(x₂−x₁)). Then use one of the points to confirm the initial value. For example, points (0, 2) and (1, 4) give a base of 4/2 = 2, so f(x) = 2 · 2^x.
Since f(0) = 2, the initial value (p) is 2. Since f(1) = 4, the base a = 4/2 = 2. Therefore the function is f(x) = 2 · 2^x = 2^(x+1).
From f(0) = 4, the initial value is p = 4. From f(1) = 12, the base a = 12/4 = 3. So the function is f(x) = 4 · 3^x.
When the base a > 1, the function f(x) = a^x increases as x increases — this is exponential growth. When 0 < a < 1, the function decreases as x increases — this is exponential decay. Real-world examples include population growth (a > 1) and radioactive decay (0 < a < 1).
Exponential functions model a vast range of real-world phenomena including compound interest, population dynamics, radioactive decay, viral spread, and signal processing. The natural exponential e^x is especially important because it appears naturally in solutions to differential equations that describe continuous growth and change.