The document discusses exponential functions and equations. It begins by explaining that exponential functions model real-life situations like population growth and decay. It then defines the exponential function f(x)=bx as having a variable exponent x and fixed base b. The graph of an exponential function either increases or decreases rapidly depending on if the exponent is positive or negative. It also shows that exponential equations set the exponents of both sides equal to solve for the variable exponent. Several examples are worked through step-by-step to demonstrate solving exponential equations algebraically by setting the exponents equal.

2.  The exponential function is very important
in math because it is used to model many
real life situations.
◦ For example: population growth and
decay, compound interest, economics,
and much more.

3. Question…
x
b
x
f 
)
(
How is this function
different from functions that
we have worked with
previously?

4. The exponent is the variable!
Answer:
x
b
x
f 
)
(
b = The base
b >0 and b ≠ 1
x = The exponent
x=any real number
Question…
Is f(x)=x3 an exponential
function?
NO

5.  One of the most common exponential
functions is
 The graph looks like this:
x
x
f 2
)
( 

6.  The graph starts off
slow but then
increases very rapidly
 The x-axis (y=0) is an
asymptote.
 X can be any real
number, for example:
 (0,1) is the y intercept
x
x
f 2
)
( 
3
2
)
( 
x
f
Models Exponential Growth

7. What would a graph look like if the
exponent was negative ?
For example:
Question…
𝑓 𝑥 = 2−𝑥

8.  The graph starts off
very high but then
decreases very rapidly
 The x-axis (y=0) is an
asymptote.
 X can be any real
number, for example:
 (0,1) is the y intercept








2
1
)
(x
f
Models Exponential Decay
𝑓 𝑥 = 2−𝑥

9. x
b
x
f 
)
(
b = The base
b >0 and b ≠ 1
x = The exponent
x=any real number
Definition
x>0 x<0

11.  What is an exponential equation?
◦ An equation where the variable is the
exponent.
◦ Example:
8
1
2 
x

12. 8
1
2 
x
Any ideas?
What if we changed
the right side to
3
2
3
2
2 

x
Now What?

13. ◦If
◦Then x=?
◦Check:
◦2-3=
3
2
2 

x
3

8
1
Then we have solved
8
1
2 
x

14.  Corresponding terms in equations are equal.
◦ What does x equal in the following:
8 1.2
= 8 𝑥
𝑥 − 6 = 25 − 6
𝑙𝑜𝑔𝑥5 = 𝑙𝑜𝑔75

15. 1.Make the bases the same (if you can)
2.Set the exponents equal and solve.
3.Check your answer
If the bases are the same, set
theexponents equal!
Main
idea

16.  Solve for x: x
x 3
2
7
7 

Are the bases equal?
YES
Set the exponents equal.
Solve for x.
x
x 3
2 
 All we have here is a simple
Algebra problem
1


x

17. )
1
(
3
2
)
1
(
7
7 



3
3
7
7 


x
x 3
2
7
7 

x = -1
It checks!

18.  Solve for x: 1
4
25
5 

 t
t
Are the bases equal?
NO
Change the right side to:
1
2
)
5
( 
t
Simplify: 2
2
4
5
5 

 t
t
Solve! 2

t

19. 1
2
2
4
25
5 


1
2
25
5 
1
4
25
5 

 t
t
t = 2
It checks!

20.  Solve each equation for x.
81 2𝑥
=
1
9
2 5𝑥−6
= 1
16
2 6
4


x

21.  Solve for x:
7
7
49 2


x
Are the bases equal?
NO
Change both sides to: 2
1
2
2
7
*
7
)
7
( 

x
Simplify: 2
3
4
2
7
7 

x
Solve!
2
3
4
2 

x
4
11

x

22. 2
3
2
4
11
7
49 

2
3
4
3
7
49 
7
7
49 2


x
x = 11/4
It checks!
2
3
4
3
2
7
)
7
( 

23. REVIEW
1. Exponential function:
x
b
x
f 
)
(
The exponent is
the variable
Key Point:
b= the base
b >0 and b ≠1
X= the exponent
X = any real number
2. Exponential Equations: An equation where the
exponent is the variable
Example:
16
2 6
4


x
How to solve:
If the bases are the same,
set the exponents equal!
4
6
4
2
2 

x
Rewrite as:
Set exponents
equal:
4
6
4 

x
Solve:
4
10

x
Check:
16
2
6
)
4
10
(
4


16
2 6
10


16
24

It checks!

24. Homework: Exponential
Equations
Worksheet