Logarithmic and Exponential Functions

Rational Exponents Review

Properties of Integer Exponents

0

m-1

1 m

n

, 0 1, 0

a 1ab , 0 a , 0

a

n nm n m n

n

mn nm

n n n m nn n

mn nn

a aa a a b a a

b b

a a

a b a a aa a

a a a

Note: power

m

n roota

Examples:

Express each exponential given in radical form and evaluate:1

2144 144 12 3

416 34 16 8

1

2100

1

2

1

100

1

100

1

10

Simplify each expression:1 24 3

5 57 142 5x y x y

1 24 3

5 57 1410x y

311

51410x y

12 4 2

3

4

9

x z

y

13 2

2 4

9

4

y

x z

13 4 2

2

9

4

y z

x

3223

2

z

x

24 63 4 x y 1

1 324 6 4x y

13 3

6 2

124 6 12

or =

x y

x y

1

2 2x y

2x

y

Applications of Rational Exponents:

A clock has a pendulum of length 99.5 cm. Determine theperiod of the pendulum to the nearest tenth of a second.

*The formula has been devised to determine the approximate relationshipbetween the period and the length of a given pendulum. This formula is derived fromthe Standard Seconds Pendulum which is about 1 meter long and has a period of 2 s.

1 1

2 22T L g

1 1

2 22T L g

1 1

2 22 0.995 9.80

2.00206591 2.0 s

* First, convert the measurement to meters. Then plug values from the problem into the given formula.

Graphing Exponential FunctionsA function that can be expressed in the form is calledan exponential function.

, 0 and 1xf x b b b

2xy 3xy

2 5xxy y The value of b determinesthe steepness of the graph.

The point (0,1) is common to the graphs.

As , 0.x y

As , x y

The range of these graphs is 0, .