Section 5.3 Exponential Functions

Exponential Functions

Definition: The exponential functions are the functions of the form f(x) = ax, where the

base a is a positive constant with a 6= 1, and x is any real number and is referred to as the

exponent or the power.

Properties of the Graphs of f(x) = ax

1. Domain is the set of all real numbers.

2. Range is the set of all positive real numbers.

3. All graphs pass through the point (0, 1).

4. The graph is continuous (no holes or jumps).

5. The graph never crosses the x-axis.

6. If a > 1, the graph is increasing (exponential growth).

7. If 0 < a < 1, the graph is decreasing (exponential decay).

Example 1: Graph f(x) = 3x, g(x) = 6x, h(x) =

✓1

3

◆x

, and k(x) =

✓1

6

◆x

.

x

y

=-

-

=

+ - -

- -

Laws of Exponents: If a and b are positive numbers and x and y are any real numbers, then

1. ax+y = axay

2. ax�y =ax

ay

3. (ax)y = axy

4. (ab)x = axbx

5. ax = ay if and only if x = y

6. For x 6= 0, ax = bx if and only if a = b

Example 2: Use the Laws of Exponents to simplify the following expressions:

a) 4�3

8�8

b)⇣

15a�5b4

5a3b�4

⌘�3

2 Fall 2019, Maya Johnson

- -

→24.23--24+3=27

→ It = 24-3=2'

→ ( 2433=212→ ( 2.334=24.34

- -

→ 3.×

= 32 ⇒ x --

2

-2

" =b× ⇒ b=Z

4-3

use abase of I

'

( 22 ,- 3

T.s-TE-EII.is ,

- 6

= 2¥ = z-

6 - ( - 247=2-61-24 , ④

- 3

¥ .

= ( 3. a-5-3

.b

4 - t -

4g- 3

=L 3. a- 8. bs )- 3

=L 3)- ? ( a

-8

)- 3. ( b8 )

-3

= 3-3 .

and 4. b- 24

24

= ;=a4

c) 4p

xy3 2p

x5y (Note: ax = ap/q = qpap = ( q

pa)p.)

d) 3

q�27a�6b2

a3b�4c�9

�

3 Fall 2019, Maya Johnson

-

-

④ JEY , ×

" 14,514

=( x y 3)

" 4. ( xsy )"

= ( ×"

. y 5)' 14

÷:=

=zTa

=/" 3

=

127 )"? ( a- 9)

"? ( blog

's

3

= 3.ci?;b=3b2ae3-J

Example 3: Solve for x:

a) 4x�123x = 84

b)1

25= 54x · 1

53x2

c) 9x�1 = 31+x

4 Fall 2019, Maya Johnson

aka"

⇒ x -- y

→ Common base

4×-123×-84 ( Common base =2 )

( 22 )" ! 23 ! ( 2314

22×-2 .

23×= 212

22×-21-34=212 ⇒ 2×-2+34=12

anm 5×-2=12

+ 2 t 2

EE- -¥ ⇒ x=I⑤→ Common base

÷ =

54×0¥5 ( C • maroon base =5 )>

- 2=4×-3×2I =54× .

1- 3×2-4×-2=0ST 53¥

⇒×=4I ⇒

" →±EEF⇒ -

~ ~

x=4T ⇒ ×=2tzor2-6

→ Common

9×-1=3 ' tx ( common base =3 )

( 34×-1=31 tx

32×-2=31TX

✓-

2x - 2 = It x

- x- ×

⇒

X -

2=1

+ 2 t2

Exponential Function with Base e Exponential functions with base e are the most commonly

used as the number “e” is a naturally occurring number in the world. The number e is approxi-

mately equal to 2.72.

Example 4: Simplifyex+5

e5�x

Example 5: Solve for x:

x2ex � 16xex = 0

Finding the Domain of Functions of the Form f(x) = eg(x)

Example 6: Find the domain of the following functions:

a) ep14 + x

5 Fall 2019, Maya Johnson

-

F--

ext 5 × +5 - 5 t X

¥= e

× +5 - CS- × ,= e

= ④

in we

x. ex ( x - 16 ) = Oin -

-

one . . + me.

.

'

>X - 16=0 ⇒

-

Domain = Domain of

gcx)

O

Find domain of TEX

Solve 14 t X 7 O

- 14- 14

X I - 14 1-17

.

⇐'

b) e

x

x+ 2

!

c)x

ex+ 2

Applications of Exponential Functions

Growth and Decay Applications

Functions of the form y = cekt,

where c and k are constants and the independent variable t represents time, are often

used to model population growth and radioactive decay. The constant c represents the

initial amount. The constant k is called the relative growth rate. We say that the

population is growing continuously at the relative growth rate of k.

Example 7: The population of a particular city grows continuously at a relative growth rate of

5.4%. If 30, 000 people currently live in the city, what will be the population in eight years?

6 Fall 2019, Maya Johnson

Dit

Find

sowext.EE

soText 2=0 Never true !

E:i

=-

-=

-

-

-

TEE-

C = 30000 p = z oooo .

54K£ )

15 -.

054 I

t --

8

Finance (Continuously Compounded Interest)

A = Pert

where P =principal, r=Annual interest rate compounded continously (as a decimal), t =Time in

years, A =Accumulated amount at the end of t years.

Example 8: What amount will an account have after five years if $1,000 is invested at an annual

rate of 3.25% compounded continuously?

Example 9: A bank note will be worth $75, 500 when it matures in 5 years. If the note pays

5.19% per year compounded continuously, determine how much an individual must pay now for

the note. (Round your answer to the nearest cent.)

7 Fall 2019, Maya Johnson

-

- -

-

A --

Pert

P -

-

tooo

A- 1000£. 032545 )

r = .0325

=$ll76.t -

- 5

- -

- - -

-

A -

.Pert

-

A -

-75500 75500 = Ped'9) (5)

r -

- . 0519¥945

⇒ Pa 75500t -

- 5

⇒ p --858,243.7€