Exponential Function Examples in Math

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

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

Concept Recap

An exponential function has the form $f(x) = a \cdot b^x$ where $b > 0$ , $b \neq 1$ . The variable is in the exponent, not the base.

Growth (or decay) that multiplies by a constant factor repeatedly.

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: An exponential function multiplies by a constant factor for every unit increase in the input.

Common stuck point: The procedure for exponential function is the easy part; the trap is putting the variable in the base instead of the exponent. Asking "Does the output multiply by the same factor for each equal step in $x$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the output multiply by the same factor for each equal step in $x$ ?

Worked Examples

Example 1

easy

A bacteria population starts at 500 and doubles every 3 hours. Write an exponential model and find the population after 9 hours.

Answer

P(9) = 4000

First step

The general form is

P(t) = P_0 \cdot b^{t/k}

where

P_0 = 500

b = 2

, and

k = 3

Full solution

2
Model: $P(t) = 500 \cdot 2^{t/3}$ .
3
At $t = 9$ : $P(9) = 500 \cdot 2^{9/3} = 500 \cdot 2^3 = 500 \cdot 8 = 4000$ .

Exponential growth models use the form

P_0 \cdot b^{t/k}

where

b > 1

represents growth. The ratio

t/k

counts how many doubling periods have elapsed.

Example 2

medium

Solve

3^{2x - 1} = 81

Example 3

medium

An investment of $1000 earns

5\%

interest compounded annually. Write a model and find the value after

4

years.

Example 4

medium

A car worth $20,000 depreciates by

15\%

per year. Write a decay model and find the value after

5

years.

Example 5

medium

The population of a town grows from

5000

6655

3

years exponentially. Find the annual growth rate.

Example 6

hard

Solve

2^{x+1} + 2^x = 24

Example 7

hard

A bacterial culture grows from

500

32{,}000

5

hours. Assuming exponential growth, find the doubling time.

Example 8

hard

A drug enters the bloodstream at

200

mg and is eliminated at

20\%

per hour. When does the amount first drop below

50

mg?

Example 9

challenge

Two species of bacteria start with the same population. Species A doubles every

4

hours, Species B triples every

9

hours. After how many hours do their populations match again, other than at

t=0

? Express the smallest positive answer.

Practice Problems

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

Example 1

medium

A radioactive substance has a half-life of 10 years. If the initial amount is 200 g, how much remains after 30 years?

Example 2

hard

Two bacteria colonies start with 100 and 500 cells respectively. Colony A doubles every 3 hours; Colony B doubles every 8 hours. After how many hours will Colony A surpass Colony B?

Example 3

easy

Evaluate

2^5

Example 4

easy

Evaluate

3^0

Example 5

easy

Evaluate

2^{-3}

Example 6

easy

f(x)=2^x

g(x)=x^2

the exponential function?

Example 7

easy

Evaluate

5^2

Example 8

easy

If a population doubles each hour starting at

100

, what is it after

3

hours?

Example 9

easy

Evaluate

\left(\frac{1}{2}\right)^3

Example 10

easy

What is the

y

-intercept of

f(x)=3\cdot 2^x

Example 11

medium

Solve

2^x=16

Example 12

medium

Solve

3^{x+1}=27

Example 13

medium

Solve

4^x=8

Example 14

medium

A quantity halves every

5

years from

80

grams. How much remains after

15

years?

Example 15

medium

Simplify

\frac{2^7}{2^3}

Example 16

medium

2^x=5

, find

2^{x+3}

Example 17

medium

Solve

2^{2x}-5\cdot 2^x+4=0

Example 18

medium

What is the horizontal asymptote of

f(x)=2^x-3

Example 19

challenge

a^x=3

and

a^y=5

, find

a^{2x+y}

Example 20

challenge

Solve

9^x=3^{x+1}

Example 21

challenge

An investment grows by

50\%

each year. After how many whole years does

\$100

first exceed

\$300

Example 22

medium

Simplify

(2^3)^2

Example 23

easy

Evaluate

f(2)

for

f(x) = 4 \cdot 3^x

Example 24

easy

f(x) = 5 \cdot 0.8^x

exponential growth or decay?

Example 25

easy

Solve

2^x = 16

Example 26

easy

A culture of

200

cells triples every day. How many cells are there after

2

days?

Example 27

medium

Solve

5^{x+1} = 125

Example 28

medium

Solve

4^x = 8

Example 29

medium

For

f(x) = 3 \cdot 2^x

, find the value of

x

such that

f(x) = 48

Example 30

medium

Solve

9^x = 27^{x-1}

Example 31

medium

For

f(x) = 5 \cdot 2^x

, compare

f(0)

and

f(3)

Example 32

medium

A radioactive sample decays at

4\%

per hour. What fraction remains after

10

hours?

Example 33

hard

Solve

3^{2x} - 4 \cdot 3^x + 3 = 0

Example 34

hard

For what value of

b

does

f(x) = b^x

pass through

(3, 64)

Example 35

hard

The function

f(x) = a \cdot b^x

passes through

(0, 5)

and

(2, 45)

. Find

a

and

b

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Related Concepts

Exponents Function Logarithm Euler's Number

Background Knowledge

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

exponentsfunction definition