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

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

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

6. f ( x)
2
x
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:
f (x) 2 3
 (0,1) is the y intercept

Models Exponential Growth

7. Question…
What would a graph look like if b is
between 0 and 1?
For example:
x
f ( x)
1
2

8. f ( x)
1
2
x



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:
f (x)

1
2
(0,1) is the y intercept
Models Exponential Decay

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

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

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

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

14. 1.Express each side of the equation as a
power of the same base.
2.Set the exponents equal and solve.
3.Check your answer
Main
idea
If the bases are
the same, set the
exponents equal!

15. 
Solve for x:
7
x 2
7
3x
Are the bases equal?
YES
Set the exponents equal.
Solve for x.
x 2 3x
All we have here is a simple
Algebra problem
x
1

16. x = -1
7
7
x 2
( 1) 2
7
3
3x
7
3( 1)
7
3
7
It checks!

17. 
Solve for x:
5
4 t
t 1
25
Are the bases equal?
NO
Change the right side to:
Simplify:
Solve!
5
4 t
t
5
2
2t 2
2 t 1
(5 )

18. t=2
4 t
t 1
5
25
4 2
2 1
5
25
2
1
5 25
It checks!

19. 
Solve for x:
49
x 2
7 7
11
x
4

20. 
Solve for x:
49
x 2
7 7
Are the bases equal?
NO
2 x
Change both sides to: (7 )
Simplify:
Solve!
7
2x 4
2
7*7
3
2
7
3
2x 4
2
11
x
4
1
2

21. x = 11/4
x 2
49 11
49 4
2
3
4
3
2 4
73 7
7 23
2
3
2
49
7
(7 )
7
It checks!

22. REVIEW
1. Exponential function:
f ( x) b
Key Point:
2. Exponential Equations:
Example:
Rewrite as:
x
The exponent is
the variable
b= the base
b >0 and b ≠1
X= the exponent
X = any real number
An equation where the
exponent is the variable
4x 6
2
16
4x 6
4
2How to solve:
2
Set exponents
4x 6 4
If the bases are the
equal:
Check:
same, set the10
exponents
x
Solve:
equal!
4
2
4(
10
) 6
4
10 6
2
2
4
16
16
16
It checks!

23. Homework:
Exponential
Equations
Worksheet