Exponential Function¶

Definition¶

Functions $f(x)=ab^x$ are called exponential functions, where $b>0$ is known as the base of the exponential function. One of the most important exponential functions is the natural exponential function $f\colon \mathbb{R}\to \mathbb{R}$ with $f(x)=\mathrm{e}^x=\exp(x)$ using exponentiation with a fixed positive base and a varying exponent; its base is Euler's number $\mathrm{e}\text{.}$

Interpretation¶

Exponential functions model processes where growth or decay is proportional to the current value. For $f(x)=ab^x$ $a b^{x}$ , increasing $x$ by $1$ multiplies the value by $b$ , since $f(x+1)=b\,f(x)$ $b f (x)$ : equal steps in $x$ give equal factors, not equal increments.

The increasing curve y = 2 to the x and the decreasing curve y = (1/2) to the x, both passing through the point (0, 1) and approaching 0 on one side.

Both curves pass through $(0,1)$ because $b^0=1$ for every base. The curve $y=2^x$ doubles with each unit step to the right, through $2$ , $4$ and $8$ at $x=1,2,3$ $1, 2, 3$ , and approaches $0$ far to the left. The curve $y=\left(\tfrac{1}{2}\right)^x=2^{-x}$ is its mirror image in the $y$ -axis. Neither curve meets the $x$ -axis, because $b^x>0$ for all $x$ .

The base must be positive for $b^x$ to be defined for every real $x$ : $(-2)^{1/2}$ is not a real number. A base $b>1$ gives growth, $0<b<1$ decay and $b=1$ a constant; the factor $a=f(0)$ is the starting value and fixes the sign. For $b\neq1$ the logarithm with base $b$ is the inverse of $b^x$ , and $b^x=\mathrm{e}^{x\ln b}$ $e^{x l n b}$ , so every exponential function is a rescaled version of $\mathrm{e}^x$ .

A common slip is to confuse $2^x$ , where the exponent varies, with the power function $x^2$ , where the base varies. The exponential eventually wins: $2^x>x^2$ for all $x>4$ $4$ , although $2^3=8<9=3^2$ $3^{2}$ .

Key Results¶

The exponential function $b^x$ is always positive; a scaled exponential function $ab^x$ therefore has the sign of $a\text{.}$
The exponential function is continuous and differentiable everywhere.
The natural exponential function is its own derivative.

Examples¶

$f(x)=2^x$ describes exponential growth.
$f(x)=\mathrm{e}^{-x}$ describes exponential decay.

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 4

Aliases: natural exponential function, exp Symbols: real numbers

Last updated: 20 October 2026 03:02