e^{x}

Exponential

Function

In plain English

Repeated multiplication by e ≈ 2.718, extended smoothly to any x. $e^x$ is the function whose growth rate equals its current value, so anything that grows (or decays) in proportion to its size follows it. $e^{-x}$ decays. Exponentials turn sums into products ( $e^{a+b} = e^a e^b$ ), are always positive, and are the inverse of the natural log. In ML, $e^{z}$ is what softmax uses to turn arbitrary scores into positive weights.

As code

Math.exp(1);        // 2.718…
Math.exp(-2);       // 0.135…  (decay)
// softmax numerator: every score becomes a positive weight
[2, 1, -1].map(Math.exp);

See it

rate1.35

e^{x}

describes growth proportional to current size. Each equal step in

x

multiplies

y

by the same factor.

What it means, field by field

Mathematics

Inverse of ln; e^{iθ} links exponentials to rotation (Euler’s formula).

Physics

Radioactive decay N₀e^{−λt}; Boltzmann factor e^{−E/k_BT}; damping.

Biology

Unconstrained population growth N₀e^{rt}; drug clearance.

AI / ML

Softmax, attention weights, the sigmoid, and learning-rate decay schedules.

In a research paper

Exponential (Malthusian) growth in population biology and microbiology, valid until resources run out.

N(t) = N_0\, e^{rt}

Each unit of time multiplies the population by the same factor e^r.

Common misreadings

Calling any fast growth "exponential". Polynomial and power-law growth are different and can look similar over short ranges.
Assuming exponential trends continue indefinitely. Real systems saturate (logistic curves).

Connected symbols

\log

Logarithm

\prod

Product (capital pi)

\frac{d}{dx}

Derivative

\sigma

Sigma (lowercase)

Where it appears on this site

∂ Scaled dot-product attention ∂ Softmax with temperature ∂ The Boltzmann distribution § 1. Learning to Read Equations § 4. Probability, Entropy, and Information § 5. Complexity and Systems Thinking § 6. The Mathematics Inside Modern AI