---
title: "The Exponential Function and the Number e"
description: "Treat y = aˣ as a family of curves: always positive, multiplied by a fixed factor per unit step, with a horizontal asymptote. One member, y = eˣ, has a gradient equal to its own value everywhere."
canonical: https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ
source: https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ.md
retrieved: 2026-09-12
---

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# The Exponential Function and the Number e

Treat y = aˣ as a family of curves: always positive, multiplied by a fixed factor per unit step, with a horizontal asymptote. One member, y = eˣ, has a gradient equal to its own value everywhere.

Subject: Mathematics · Area: Algebra · Ages 16 to 17
Page: https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ

## Ready when they can

- Describe the shape and asymptote of y = 2ˣ and of y = 2⁻ˣ
- Say what makes e different from any other base
- Use a model of the form A e^(kt) to describe growth and decay

## Lesson: Growth that multiplies every step

Each step multiplies by the same base. With base 2 the values run 1, 2, 4, 8. Every curve stays positive and flattens toward the x axis on the left. That floor line is the asymptote y equals 0.

**Example.** Compare y equals 2 to the x with y equals 2 to the minus x. The first climbs to the right, the second decays to the right. Flipping the sign of x mirrors the graph. Both share the same asymptote.

One base is special. The number e is about 2.718, and y equals e to the x has a gradient equal to its own value everywhere. At the y axis crossing the gradient is 1. That property is what makes e different from every other base.

**Tip.** The model A e to the k t fits growth and decay in proportion to size. A sets the starting amount and k sets the pace. A positive k climbs, a negative k falls toward zero. Spotting A and k in a story is half the battle.

**Recap.** Fixed factor per step, asymptote at zero, and e as the base whose gradient matches its value.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Recognising Cubic, Reciprocal and Exponential Graphs](https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9)
- [Compound Growth and Decay](https://lightmysky.com/learn/mathematics/compound-growth-and-decay-mt_xf5FQ2sugf)

## Opens up

- [Derivatives of Exponential, Logarithmic and Trigonometric Functions](https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT)
- [Hyperbolic Functions and Their Inverses](https://lightmysky.com/learn/mathematics/hyperbolic-functions-and-their-inverses-mt_gCUl1sWi9M)
- [Euler's Formula and the Exponential Form](https://lightmysky.com/learn/mathematics/eulers-formula-and-the-exponential-form-mt_RX2IxsnSwG)
- [Logarithms as the Inverse of Exponentials](https://lightmysky.com/learn/mathematics/logarithms-as-the-inverse-of-exponentials-mt_skLQLUan0T)
- [The Boltzmann Factor and the Partition Function](https://lightmysky.com/learn/science/the-boltzmann-factor-and-the-partition-function-mt_u9MSoTgJkF)