Euler's Number Examples: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Euler's Number.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Euler's number $e \approx 2.71828$ is the unique base for which the exponential function $e^x$ is its own derivative — the natural base for growth and decay.

The 'natural' base for growth—what you get from continuous compounding.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Euler's number $e\approx 2.718$ is the base whose exponential grows at a rate equal to its own value at every instant.

Common stuck point: The procedure for euler's number is the easy part; the trap is treating $e$ as a variable to solve for. Asking "Is the growth happening continuously, with the natural base where the rate equals the amount?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the growth happening continuously, with the natural base where the rate equals the amount?

Worked Examples

Example 1

easy

Evaluate

\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n

and state what this limit defines.

Answer

e \approx 2.71828

First step

Substitute increasing values of

n

: for

n=10

\left(1+\frac{1}{10}\right)^{10} \approx 2.5937

; for

n=100

\approx 2.7048

; for

n=1000

\approx 2.7169

Full solution

2
Observe that as $n \to \infty$ the expression approaches a fixed value that does not grow without bound.
3
By definition, $\lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n = e \approx 2.71828\ldots$ , Euler's number.

Euler's number

e

is defined as this limit. It is irrational and transcendental, appearing naturally in continuous growth, calculus, and complex analysis.

Example 2

hard

Show that the derivative of

f(x) = e^x

is itself, i.e.,

f'(x) = e^x

, using the limit definition of the derivative.

Example 3

medium

Simplify

\ln(e^7)

Example 4

medium

Carbon-14 decay:

A(t) = A_0 e^{-0.000121 t}

years. After

5730

years, what fraction remains? Use

e^{-0.693}\approx 0.5

Example 5

medium

Why does continuous compounding cap the limit of

(1+r/n)^{nt}

e^{rt}

n\to\infty

Example 6

hard

Compare

e^\pi

and

\pi^e

. Use

e^\pi \approx 23.14

and

\pi^e \approx 22.46

Example 7

challenge

Using the series

e^x = \sum_{k=0}^{\infty} \tfrac{x^k}{k!}

, approximate

e

using the first 5 terms (

k=0

4

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Using a calculator, compute

e^2

to four decimal places. Then determine whether

e^2 > 7

Example 2

medium

A bank offers continuous compounding at an annual rate of

5\%

. Using

A = Pe^{rt}

, find how much

\$1000

grows to after

10

years.

Example 3

easy

e

a variable or a constant?

Example 4

easy

Approximate

e^2

using

e\approx 2.718

. (one decimal)

Example 5

easy

Which is larger:

e

\pi

Example 6

easy

What is

e^0

Example 7

easy

The natural base

e

arises from what kind of process?

Example 8

easy

Approximate

e^1

to two decimals.

Example 9

easy

n\to\infty

, what does

\left(1+\frac{1}{n}\right)^n

approach?

Example 10

easy

Distinguish

e^x

from

xe

x=2

using

e\approx 2.718

Example 11

medium

A population grows as

P(t)=100e^{0.5t}

. Find

P(2)

using

e\approx 2.718

Example 12

medium

Simplify

e^3 \cdot e^2

to a single power of

e

Example 13

medium

Simplify

\dfrac{e^7}{e^4}

Example 14

medium

Radioactive decay:

A(t)=80e^{-0.1t}

. Find

A(0)

and state the long-run limit.

Example 15

medium

Solve

e^x = 1

for

x

Example 16

medium

Order from smallest to largest:

e^{-1}

e^0

e^1

. Use

e\approx 2.718

Example 17

medium

Continuous growth at rate

r=0.03

for

t=10

multiplies an amount by what factor? Use

e^{0.3}\approx 1.3499

Example 18

medium

Why does

e^x

have the special property among exponentials that its slope at any point equals its value?

Example 19

medium

Simplify

(e^2)^3

to a single power of

e

Example 20

challenge

Solve

100e^{0.04t}=200

for

t

. Use

\ln 2\approx 0.693

Example 21

challenge

Using

(1+1/n)^n

, estimate

e

n=2

and explain why it underestimates

e

Example 22

challenge

\ln(A)=\ln(B)+3

, express

A

in terms of

B

Example 23

easy

Approximate

e^{-1}

to three decimal places using

e \approx 2.71828

Example 24

easy

Approximate

(1+1/4)^4

as an estimate of

e

. Round to three decimals.

Example 25

easy

Solve

e^x = e^5

for

x

Example 26

easy

Which is larger,

e^2

7

? Use

e \approx 2.718

Example 27

medium

Continuous compounding: $500 at

4\%

annual rate for

6

years. Find the amount. Use

e^{0.24}\approx 1.27125

Example 28

medium

A bacterial culture follows

N(t) = 200 e^{0.3t}

(hours). What is

N(0)

and the doubling time? Use

\ln 2 \approx 0.693

Example 29

medium

Simplify

e^{\ln 5}

Example 30

medium

Solve

e^{2x} = 9

. Express

x

using

\ln

Example 31

medium

Find the slope of

f(x)=e^x

x=0

Example 32

medium

Evaluate

\int_0^1 e^x \, dx

. Use

e \approx 2.718

Example 33

medium

Simplify

\dfrac{e^{2x}\cdot e^{3}}{e^{x}}

as a single power of

e

Example 34

medium

A car loses value continuously:

V(t)=20000 e^{-0.15 t}

dollars. Find

V(5)

. Use

e^{-0.75}\approx 0.4724

Example 35

hard

Solve

3 e^{2x} = 24

for

x

. Use

\ln 2 \approx 0.693

Example 36

hard

Find the equation of the tangent line to

y=e^x

x=1

. Use

e \approx 2.718

Example 37

hard

Solve

e^{x} = 2 e^{-x}

for

x

. Use

\ln 2 \approx 0.693

Example 38

hard

Compute

\lim_{x \to 0}\dfrac{e^x - 1}{x}

Example 39

challenge

A population doubles every 5 years and grows continuously:

P(t)=P_0 e^{rt}

. Find

r

using

\ln 2\approx 0.693

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Related Concepts

Exponential Function Natural Logarithm

Background Knowledge

These ideas may be useful before you work through the harder examples.

exponential function