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The Number e and eˣ

Q: What is the number e?

An irrational constant, about 2.718, that arises naturally from continuous growth. It is the base of the natural exponential e^x and the natural logarithm.

Q: Why is e^x so important?

Because it is the unique exponential whose gradient equals its value everywhere — its own derivative. That makes it the natural model for anything growing or decaying at a rate proportional to its current size.

Q: How do I tell growth from decay in Ae^kt?

Look at the sign of k. A positive k means growth (the value increases); a negative k means decay (it falls towards zero). A is the value at t = 0.

Q: What is the natural logarithm?

x is _e x, the inverse of e^x. It undoes exponentiation to base e: (e^x) = x. It is the tool for solving equations involving e.

Among all exponential functions, one is special: $y = e^{x}$ , where $e \approx 2.718$ . It is the unique curve whose gradient at every point equals its own height — which is exactly why calculus, and every natural growth model, is built around it.

24 min Video by TLMaths Exponentials and Logarithms

The big picture

Why should mathematics single out an awkward number like $2.718 \dots$ ? Because $e$ is the base for which growth rate exactly equals current size. A population whose growth is proportional to its size, money compounding continuously, radioactive decay — all are governed by $e^{k x}$ . The deep reason is that $e^{x}$ is the one exponential that is $its own derivative$ : the rate at which it changes is itself. That property makes $e$ the natural language of change, and it is the bridge from this chapter into the whole of calculus with exponentials.

A-Level Maths F2-01: Another way of deriving eWatch the walkthrough, then work through the notes below.

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What you'll be able to do

Know that $e \approx 2.718$ and recognise $y = e^{x}$ as an exponential
State the special gradient property of $e^{x}$
Use $e^{k x}$ in growth and decay models
Relate $e^{x}$ and its inverse, the natural logarithm $ln x$

A very special base

$e$ is an irrational constant, roughly $2.71828$ , that arises naturally from continuous growth (for instance, compound interest compounded infinitely often approaches an $e$ factor). It sits between the bases 2 and 3, so the graph of $y = e^{x}$ lies between $y = 2^{x}$ and $y = 3^{x}$ — same shape, through $(0, 1)$ , asymptote $y = 0$ .

What sets $e$ apart is not its graph’s appearance but a property you will prove with calculus.

Why this works

Like $π$ , $e$ is a constant nature keeps handing us. $π$ shows up wherever there are circles; $e$ shows up wherever there is growth proportional to size.

The property that makes e matter

The function $y = e^{x}$ is the $only$ exponential whose $gradient equals its value$ at every point. Where the curve is at height 5, its steepness is also 5. In the language of the next chapter, $e^{x}$ is its own derivative.

This is why $e$ dominates growth models: “rate of change proportional to current amount” is the definition of natural growth, and $e^{k x}$ satisfies it exactly, with $k$ controlling the speed (positive for growth, negative for decay).

\frac{d}{d x} (e^{x}) = e^{x}

The gradient of

e^{x}

e^{x}

— unique among exponentials.

1At

t = 0

y = 100 e^{0} = 100 \times 1 = 100

, the initial amount.

2The exponent

0.2 t

has a positive coefficient, so it grows.

AnswerInitial amount 100; growing (positive exponent coefficient).

Tip — $e^{k x}$ grows when $k > 0$ and decays when $k < 0$ . The size of $k$ sets how fast.

Its inverse: the natural logarithm

Since $e^{x}$ is a one-to-one function, it has an inverse — the $natural logarithm$ $ln x$ (that is $lo g_{e} x$ ). It undoes $e^{x}$ : $ln (e^{x}) = x$ and $e^{l n x} = x$ . This pairing is how you will solve equations like $e^{x} = 5$ (take $ln$ of both sides), covered fully in the logarithm lessons.

ln x = lo g_{e} x, ln (e^{x}) = x

The natural log is the inverse of

e^{x}

Think like an examiner

Treat $e$ as a specific number ( $\approx 2.718$ ), not a variable — use the $e^{x}$ button on your calculator.
In a model $A e^{k t}$ , the initial value is $A$ (at $t = 0$ ) and the sign of $k$ tells you growth or decay.
Pair $e^{x}$ with $ln$ : to “undo” an $e$ , take natural logs; to undo a $ln$ , exponentiate.
Leave answers exact in terms of $e$ or $ln$ when the question asks for exact form.

Common misconceptions

Treating

e

as an unknown to be solved for.

e

is a fixed constant, about 2.718 — like

π

Thinking

\frac{d}{d x} (e^{x}) = x e^{x - 1}

(power-rule style).

e^{x}

is its own derivative:

\frac{d}{d x} (e^{x}) = e^{x}

Reading

100 e^{0.2 t}

as a starting value of

100 e

t = 0

e^{0} = 1

, so the initial value is 100.

Forgetting

ln

is the inverse of

e^{x}

ln (e^{x}) = x

and

e^{l n x} = x

— they cancel.

e essentials

e \approx 2.71828

The natural base.

\frac{d}{d x} (e^{x}) = e^{x}

Its own gradient.

y = A e^{k t}

Growth (

k > 0

) / decay (

k < 0

ln x = lo g_{e} x

The inverse function.

Stretch yourself

The mass of a radioactive sample is $M = 80 e^{- 0.05 t}$ grams, with $t$ in years. Find the initial mass and the mass after 20 years, and state what happens in the long term.

Hint — Initial mass is at $t = 0$ . Substitute $t = 20$ for the later mass. Consider $e^{- 0.05 t}$ as $t \to \infty$ .

Questions students ask

What is the number e?

An irrational constant, about 2.718, that arises naturally from continuous growth. It is the base of the natural exponential

e^{x}

and the natural logarithm

ln

Why is

e^{x}

so important?

Because it is the unique exponential whose gradient equals its value everywhere — its own derivative. That makes it the natural model for anything growing or decaying at a rate proportional to its current size.

How do I tell growth from decay in

A e^{k t}

Look at the sign of

k

. A positive

k

means growth (the value increases); a negative

k

means decay (it falls towards zero).

A

is the value at

t = 0

What is the natural logarithm?

ln x

lo g_{e} x

, the inverse of

e^{x}

. It undoes exponentiation to base

e

ln (e^{x}) = x

. It is the tool for solving equations involving

e

Key takeaways

$e \approx 2.718$ is the natural exponential base.
$y = e^{x}$ is the unique exponential equal to its own gradient.
$A e^{k t}$ models growth ( $k > 0$ ) and decay ( $k < 0$ ); $A$ is the initial value.
The natural log $ln x = lo g_{e} x$ is the inverse of $e^{x}$ .
This property makes $e$ central to all calculus of growth.

How this fits the course

Build on

Exponential Functions

Logarithms
Differentiation (later)soon

Leads to

Logarithms

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