Section 5-1

Growth and Decay: Integral Exponents

Exponential Growth and Decay

Look at examples and graphs on page 169-170

Exponential Growth and Decay Formula

Where is the initial amount, the amount at time t = 0, and r is the growth rate (the percent changed to a decimal) and t is the time. If r > 0, then the initial amount grows exponentially. If -1 < r < 0, then the initial amount decays exponentially.

decay - )1()(

growth - )1()(

0

0

t

t

rAtA

rAtA

0A

Laws of Exponents

Same Bases:

1.

2.

3. If b ≠ 0, 1, or -1, then if and only if x = y.

yxyx bbb 0)(b yx

y

x

bb

b

yx bb

Laws of Exponents

Same Exponents:

4.

5.

6. If x ≠ 0, a > 0, and b > 0, then if and only if a = b.

xxx baab

0b

x

xx

b

a

b

a

xx ba

Laws of Exponents

Power of a Power:

7. xyyx bb

Definition of

If Law 1 is to hold for y = 0, then we have

. Since behaves like the number 1, we define it to be 1:

=1 (b ≠ 0)

0b

xxx bbbb 00 0b

0b

Definition of

If Law 1 is to hold for y = -x and b ≠ 0, then we have . Since

and have a product of 1, they are reciprocals of each other. Therefore we define: (x > 0 and b ≠ 0)

xb

10 bbbb xxxx xbxb

x

x

bb

1

Additional Example:

In a certain city, the value of a house is increasing at a rate of 16% annually.

Find the value of a $100,000 house in 4 years In how many years will the value of the house

be approximately double what it is now?

Example

Simplify each expression.

1

63 63

a

aa1

63 63

a

aa

Example

Simplify each expression.

2

63 105

b

bb2

63 105

b

bb

Additional Example:

Simplify by using powers of the same base.

a. b. 1

58

16

42

n

n

2

3

125

525