2. Concepts & Objectives
Exponential Functions
Use exponent properties to solve equations
Substitute values into exponential functions
3. Properties of Exponents
Recall that for a variable x and integers a and b:
a b a b
x x x
a
a b
b
x
x
x
b
a ab
x x
b
a b a
x x
a b
x x a b
5. Simplifying Exponents
Example: Simplify
1.
2.
3.
3 2 2
2 5
25
5
x y z
xy z
4
2 3
2r s t
2 3 1 2
5 6
y y
3 1 2 2 2 5
5x y z
4 2 4 3
4 4
2 r s t
2 1
3 2
5 6 y
2 4 3
5x y z
8 12 4
16r s t
1
6
30y
6. Exponential Functions
If a > 0 and a 1, then
defines the exponential function with base a.
Example: Graph
Domain: –∞, ∞
Range: 0, ∞
y-intercept: 0, 1
x
f x a
2x
f x
7. Exponential Functions (cont.)
Characteristics of the graph of :
1. The points are on the graph.
2. If a > 1, then f is an increasing function; if 0 < a < 1, then
f is a decreasing function.
3. The x-axis is a horizontal asymptote.
4. The domain is –∞, ∞, and the range is 0, ∞.
x
f x a
1
1, , 0,1 , 1,a
a
8. Exponential Equations
Exponential equations are equations with variables as
exponents.
If you can re-write each side of the equation using a
common base, then you can set the exponents equal to
each other and solve for the variable.
Example: Solve
1
5
125
x
3
5 5
x
3
x
3
125 5
10. Exponential Equations (cont.)
Example: Solve
1 3
3 9
x x
2
9 3
3
1 2
3 3
x
x
1 2 3
x x
1 2 6
x x
7 x
2 3
1
3 3 x
x
11. Exponential Equations (cont.)
To solve an equation with exponents, you can “undo” the
exponent by raising each side to the reciprocal.
Solve
5 2
243
b
12. Exponential Equations (cont.)
To solve an equation with exponents, you can “undo” the
exponent by raising each side to the reciprocal.
Solve
5 2
243
b
2
2
5 5
5
2
243
b
9
b
13. Compound Interest
The formula for compound interest (interest paid on
both principal and interest) is an important application
of exponential functions.
Recall that the formula for simple interest, I = Prt, where
P is principal (amount deposited), r is annual rate of
interest, and t is time in years.
14. Compound Interest (cont.)
Now, suppose we deposit $1000 at 10% annual interest.
At the end of the first year, we have
so our account now has 1000 + .11000 = $1100.
At the end of the second year, we have
so our account now has 1100 + .11100 = $1210.
1000 0.1 100
I
1100 .1 110
I
15. Compound Interest (cont.)
Another way to write 1000 + .11000 is
After the second year, this gives us
1000 1 .1 .1 1000 1 .1
1000 1 .1 1 .1
2
1000 1 .1
1000 1 .1
16. Compound Interest (cont.)
If we continue, we end up with
This leads us to the general formula.
Year Account
1 $1100 10001 + .1
2 $1210 10001 + .12
3 $1331 10001 + .13
4 $1464.10 10001 + .14
t 10001 + .1t
17. Compound Interest Formulas
For interest compounded n times per year:
For interest compounded continuously:
where e is the irrational constant 2.718281…
1
tn
r
A P
n
rt
A Pe
18. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
19. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
2 10
.06
2500 1
2
A P = 2500, r = .06,
n = 2, t = 10
20. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
2 10
.06
2500 1
2
A
21. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
2 10
.06
2500 1
2
A = $4515.28
22. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
2 10
.06
2500 1
2
A
4 8
.048
15000 1
4
P A = 15000, r = .048,
n = 4, t = 8
= $4515.28
23. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
2 10
.06
2500 1
2
A
4 8
.048
15000 1
4
P
15000 1.4648
P
= $4515.28
24. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
2 10
.06
2500 1
2
A
4 8
.048
15000 1
4
P
15000 1.4648
P
= $4515.28
25. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
2 10
.06
2500 1
2
A
4 8
.048
15000 1
4
P
15000 1.4648
P
$10,240.35
P
= $4515.28
26. Examples
3. If $8000 is deposited in an account paying 5% interest
compounded continuously, how much is the account
worth at the end of 6 years?
4. Which is a better deal, depositing $7000 at 6.25%
compounded every month for 5 years or 5.75%
compounded continuously for 6 years?
27. Examples
3. If $8000 is deposited in an account paying 5% interest
compounded continuously, how much is the account
worth at the end of 6 years?
4. Which is a better deal, depositing $7000 at 6.25%
compounded every month for 5 years or 5.75%
compounded continuously for 6 years?
.05 6
8000
A e
$10,798.87
A
12 5
.0625
7000 1
12
$9560.11
A
.0575 6
7000
$9883.93
A e