Exponential Growth Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Exponential Growth.

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

Concept Recap

Exponential growth occurs when a quantity increases by a constant multiplicative factor over equal intervals.

Exponential growth means the amount added each period is proportional to the current amount — the bigger it gets, the faster it grows, creating an accelerating curve.

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: Exponential growth multiplies by a constant factor each period, so the bigger it gets the faster it grows.

Common stuck point: The procedure for exponential growth is the easy part; the trap is modeling percent growth as linear (adding a fixed amount). Asking "Is the quantity multiplied by the same factor each period (rather than having a fixed amount added)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the quantity multiplied by the same factor each period (rather than having a fixed amount added)?

Worked Examples

Example 1

easy

A population of bacteria doubles every

3

hours. If there are initially

500

bacteria, how many will there be after

12

hours?

Answer

8{,}000 \text{ bacteria}

First step

The exponential growth model is

P(t) = P_0 \cdot 2^{t/d}

, where

d

is the doubling time.

Full solution

2
Substitute: $P(12) = 500 \cdot 2^{12/3} = 500 \cdot 2^4$ .
3
$P(12) = 500 \cdot 16 = 8{,}000$ .

Exponential growth occurs when a quantity increases by a constant percentage over equal time intervals. The doubling-time formula

P(t) = P_0 \cdot 2^{t/d}

is a special case of

P(t) = P_0 \cdot b^t

. In 12 hours, the population doubles 4 times:

500 \to 1000 \to 2000 \to 4000 \to 8000

Example 2

medium

A city's population grows at

3\%

per year. If the current population is

200{,}000

, when will it reach

300{,}000

Example 3

medium

A town of

25{,}000

grows by

4\%

per year. Find the population after

10

years.

Example 4

medium

The number of users of an app grows by

15\%

each month. Currently there are

8000

users. After how many months will the app have

20{,}000

users?

Example 5

medium

A radioactive sample doubles every

5

years (consider a strange isotope used in modeling). If

200

g now, find an expression for its mass and evaluate at

t = 25

Example 6

medium

Compare $1000 at

6\%

compounded annually vs

\$1000

5.9\%

compounded continuously after

10

years.

Example 7

hard

At what continuous interest rate must $5000 be invested to double in

9

years?

Example 8

hard

A virus spreads in a population so that every infected person infects

1.4

more per day on average. Starting with

50

cases, how many cases on day

7

, assuming pure exponential growth where total cases multiply by

1.4

daily?

Example 9

challenge

A starting population of

1

doubles every minute and fills a jar at minute

60

. At what minute is the jar half full?

Practice Problems

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

Example 1

medium

An investment grows from

\$1{,}000

\$1{,}500

5

years with continuous compounding. Find the annual growth rate.

Example 2

hard

Two populations grow exponentially:

P_1(t) = 100 \cdot 3^t

and

P_2(t) = 5000 \cdot (1.2)^t

. When will

P_1

overtake

P_2

Example 3

easy

Evaluate

y = 3\cdot 2^x

x = 0

Example 4

easy

Evaluate

y = 5\cdot 2^x

x = 3

Example 5

easy

y = 4\cdot 3^x

growth or decay?

Example 6

easy

A population doubles each year starting at

100

. Write its model.

Example 7

easy

5\%

annual growth rate gives what base

b

Example 8

easy

Evaluate

y = 2\cdot 10^x

x = 2

Example 9

easy

Does

y = 7\cdot \left(\frac{1}{2}\right)^x

grow or decay?

Example 10

easy

What is the initial value of

y = 8\cdot 1.2^x

Example 11

medium

A bacteria culture of

200

triples every hour. How many after

3

hours?

Example 12

medium

An investment of

\$1000

grows

10\%

per year. Find its value after

2

years.

Example 13

medium

A quantity grows from

50

200

over two equal steps. Find the constant multiplier.

Example 14

medium

How long until

y = 100\cdot 2^x

first exceeds

1000

? (integer

x

)

Example 15

medium

A town of

8000

shrinks

25\%

per year. Write its model and value after 1 year.

Example 16

medium

Compare

y = 2x

and

y = 2^x

x = 10

Example 17

medium

y = a\cdot b^x

passes through

(0,5)

and

(1,15)

, find

a

and

b

Example 18

medium

A culture grows

8\%

per hour. By what factor does it grow in a full day (express as a power)?

Example 19

challenge

A model is

A = 500\cdot 2^{t/3}

with

t

in years. Find the value at

t = 9

and state the doubling time.

Example 20

medium

A car worth

\$20000

loses

20\%

of its value each year. Find its value after 1 year.

Example 21

challenge

Two accounts start at

\$1000

: one earns

\$100

per year (simple), one earns

10\%

per year (compound). After 2 years, which is larger and by how much?

Example 22

challenge

A quantity is

y = a\cdot b^x

. It grows from

80

x=1

320

x=3

. Find

b

Example 23

easy

Evaluate

y = 6 \cdot 2^x

x = 4

Example 24

easy

A quantity grows at

8\%

per year. What is the growth factor

b

Example 25

easy

Evaluate

y = 2 \cdot 3^x

x = 2

Example 26

easy

A culture of

200

cells triples every hour. Write a model for the count after

t

hours.

Example 27

medium

$2000 is invested at

5\%

compounded annually. What is the balance after

8

years?

Example 28

medium

A bacterial colony doubles every

4

hours. If it starts at

300

, how many bacteria are there after

20

hours?

Example 29

medium

A car's value depreciates by

12\%

per year. It was purchased for $28{,}000. What is its value after

5

years?

Example 30

medium

f(x) = 100 \cdot 2^x

and

g(x) = 100 \cdot 4^x

, what is

g(x)/f(x)

Example 31

medium

Solve

5 \cdot 3^x = 405

for

x

Example 32

medium

A YouTube channel grows from

1000

6000

subscribers in

4

years. Assuming exponential growth, find the annual growth rate.

Example 33

hard

A population grows continuously according to

P(t) = P_0 e^{0.04 t}

. How long until the population triples?

Example 34

hard

A bank account earns

4\%

compounded quarterly. Find the effective annual rate.

Example 35

hard

Solve

2^{x+1} = 3 \cdot 5^x

for

x

, to two decimals.

Example 36

hard

A model predicts

P(t) = 1200 \cdot (1.07)^t

. Estimate the doubling time.

Example 37

hard

Population

P_1(t) = 500 \cdot (1.1)^t

and

P_2(t) = 200 \cdot (1.18)^t

both grow. After how many years are they equal?

Example 38

challenge

Suppose

f(x) = a \cdot b^x

passes through

(1, 6)

and

(4, 162)

. Find

a

and

b

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Related Concepts

Exponential Function Growth vs Decay Compound Interest

Background Knowledge

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

exponential functiongrowth vs decaycompound interest