Math130 ch09

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Exponential and Logarithmic Functions

Chapter 9

Angel, Intermediate Algebra, 7ed 2

9.1 – Composite and Inverse Functions

9.2 – Exponential Functions

9.3 – Logarithmic Functions

9.4 – Properties of Logarithms

9.5 – Common Logarithms

9.6 – Exponential and Logarithmic Equations

9.7 – Natural Exponential and Natural Logarithmic Functions

Chapter Sections

§ 9.1

Composite and Inverse Functions

Composite Functions

The composite function is defined as

.( )( ) [ ( )] f g x f g xf g

Example:

Given f(x) = x2 – 3, and g(x) = x + 2, find .( )( )f g x

2( )( ) ( ) ( ) 3f g x f g x g x

2( 2) 3x 2 4 1x x x + 2 is substituted into each x in f(x).

2( 4 4) 3x x

g(x) is substituted into each x in f(x).

Composite Functions

Example:

Given f(x) = x2 – 3, and g(x) = x + 2, find .( )( )g f x

( )( ) ( ) ( ) 2g f x g f x f x

2 3 2x 2 1x

x2 - 3 is substituted into each x in g(x).

f(x) is substituted into each x in g(x).

One-to-One Functions

For a function to be one-to-one, it must not only pass the vertical line test, but also the horizontal line test.

A function is a one-to-one function if each value in the range corresponds with exactly one value in the domain.

Function

Not a one-to-one function

One-to one function

Inverse Functions

Function: {(2, 6), (5,4), (0, 12), (4, 1)}

If f(x) is a one-to-one function with ordered pairs of the form (x,y), its inverse function, f -1(x), is a one-to-one function with ordered pairs of the form (y,x).

Inverse Function: {(6, 2), (4,5), (12, 0), (1, 4)}

• Only one-to-one functions have inverse functions.

• Note that the domain of the function becomes the range of the inverse function, and the range becomes the domain of the inverse function.

Inverse Functions

1. Replace f(x) with y.

2. Interchange the two variables x and y.

3. Solve the equation for y.

4. Replace y with f –1(x). (This gives the inverse function using inverse function notation.)

To Find the Inverse Function of a One-to-One Function

Example:

Find the inverse function of

Graph f(x) and f(x) –1 on the same axes.

.1, 1f x x x

Inverse Functions

1, 1f x x

22 1x y

2 1x y

1 2( ) 1, 0f x x x

Replace f(x) with y.

Interchange x and y.

Solve for y.

Replace y with f –1(x) .

Example continued:

Inverse Functions

1, 1f x x x

1 2( ) 1, 0f x x x

Note that the symmetry is about the line y = x.

1 2( )( ) 1 1f f x x

1f x x

Composites and Inverses

If two functions f(x) and f –1(x) are inverses of each other, . 1 1( )( ) and ( )( )f f x x f f x x

Example:

Show that .

1, 1f x x x 1 2( ) 1, 0f x x x and

.1 1( )( ) and ( )( ) f f x x f f x x

1 2( ) 1f x x

21( ) 1 1f x x

1 1x x

§ 9.2

Exponential Functions

For any real number a > 0 and a 1,

f(x) = ax

is an exponential function.

For all exponential functions of this form,

1. The domain of the function is

2. The range of the function is

3. The graph passes through the points

.( , ) .(0, )

.1( 1, ), 0,1 , 1,aa

Exponential Graphs

Example:

Graph the function f(x) = 3x.

Domain: Range: {y|y > 0}

Exponential Graphs

Domain: Range: {y|y > 0} -1, 3

0,1 11,3

Notice that each graph passes through the point (0, 1).

Example:

Graph the function f(x) = 1 .3

§ 9.3

Logarithmic Functions

For all positive numbers a, where a 1,

y = logax means x = ay.

y = logax

logarithm(exponent)

number

means x = ay

number base

exponent

Exponential Form Logarithmic Form

50 = 1 log101= 0

23 = 8 log28= 3

41 1=2 16 1 2

1log = 416

-2 16 =36 6

1log = -236

Logarithmic Functions

For all logarithmic functions of the form y = logax or f(x) = logax, where a > 0, a 1, and x > 0,

1. The domain of the function is .

2. The range of the function is .

3. The graph passes through the points

1( , 1), 1,0 , ,1 .aa

Logarithmic Graphs

Range: Domain: {x|x > 0}

Graph the function f(x) = log10x.

1 , -110

Notice that the graph passes through the point (1,0).

Example:

Exponential vs. Logarithmic Graphs

Exponential Function Logarithmic Functiony = ax (a > 0, a 1)

y = logax (a > 0, a 1)

1 , 1a

Domain:

Range:

Points on Graph: x becomes y

y becomes x

f(x) = log10x

f(x) = 10x

Notice that the two graphs are inverse functions.

f -1(x)

Exponential vs. Logarithmic Graphs

§ 9.4

Properties of Logarithms

Product Rule

For positive real numbers x, y, and a, a 1,Product Rule for Logarithms

Example:

log5(4 · 7) = log54 + log57

log10(100 · 1000) = log10100 + log101000 = 2 + 3 = 5

log log loga a axy x y

Quotient Rule

For positive real numbers x, y, and a, a 1,Quotient Rule for Logarithms

Example:

log log loga a ax x yy

7 7 710log log 10 log 22

10 10 101log log 1 log 1000 0 3 3

Property 1

Power Rule

If x and y are positive real numbers, a 1, and n is any real number, then

Power Rule for Logarithms

Example:

log logna ax n x

49 9log 3 4log 3

210 10log 100 2log 100 2 2 4

Property 2

Additional Properties

If a > 0, and a 1, Additional Properties of Logarithms

Example:

log xa a x

49log 9 4

610log 10 6

Property 5

Property 4log ( 0)xaa x x

Combination of Properties

Example:

6 6 65log ( 3) [2 log ( 4) 3log ]x x x

Write the following as the logarithm of a single expression.

5 2 36 6 6log ( 3) [log ( 4) log ]x x x Power Rule

5 2 36 6log ( 3) [log ( 4) ]x x x Product Rule

6 2 3( 3)log

Quotient Rule

§ 9.5

Common Logarithms

The common logarithm of a positive real number is the exponent to which the base 10 is raised to obtain the number.

If log N = L, then 10L = N.

The antilogarithm is the same thing as the inverse logarithm.

If log N = L, then N = antilog L.

log 962 = 2.98318

Number Exponent

antilog 2.98318 = 962

NumberExponent

Example:

§ 9.6

Exponential and Logarithmic Equations

Properties

a. If x = y, ax = ay.

b. If ax = ay, then x = y.

c. If x = y, then logbx = logby (x > 0, y > 0).

d. If logbx = logby, then x = y (x > 0, y > 0).

Properties for Solving Exponential and Logarithmic Equations

Properties 6a-6d

Solving Equations

Example:

Solve the equation 4 256.x

2 82 2x Rewrite each side with the same base.

2 82 2x

2 8x Property 6b.

4x Solve for x.

Example:

Solve the equation log( 3) log log 4.x x Product Rule

Property 6d.

log( 3) log 4x x

( 3) 4x x 2 3 4x x

2 3 4 0x x

( 4)( 1) 0x x

4 or 1x x

Check:

Stop! Logs of negative numbers are not real numbers.

log( 3) lo4 4g( ) log 4.

log( 3) log( ) l1 1 og 4. log 4 0 log 4

log 4 log 4 True

Solving Equations

§ 9.7

Natural Exponential and Natural Logarithmic

Functions

Definitions

The natural exponent function is

f(x) = ex

where e 2.71823.

Natural logarithms are logarithms to the base e. Natural logarithms are indicated by the letters ln.

logex = ln x

Example:

ln 1 = 0 (e0 = 1) ln e = 1 (e1 = e)

Change of Base Formula

For any logarithm bases a and b, and positive number x,Change of Base Formula

loglog

This is very useful because common logs or natural log can be found using a calculator.

Example:

3log 198 log198log3

2.29670.4771

4.8136 Note that the natural log could have also been used.ln198

ln 3 5.2883

1.0986 4.8136

Properties

Notice that these are the same properties as those for the common logarithms.

Properties for Natural Logarithmsln ln ln ( 0 and 0)

ln ln ln ( 0 and 0)

ln ln ( 0)n

xy x y x y

x x y x yy

x n x x

Product Rule

Power Rule

Quotient Rule

Additional Properties for Natural Logarithms and Natural Exponential Expressions

Property 7

Property 8ln

Solving Equations

Example: Solve the following equation.

ln( 3) ln( 3) ln 40x x

ln[( 3)( 3)] ln 40x x Product Rule

2ln( 9) ln 40x Simplify

2 9 40x Property 6d

2 49x Solve for x.

7x Check solutions in original equation. (You will notice that only the positive 7 yields a true statement.)

Applications

In 2000, a lake had 300 trout. The growth in the number of trout is estimated by the function g(t) = 300e0.07t where t is the number of years after 2000. How many trout will be in the lake in a) 2003? b) 2010?

In the year 2000, t = 0. (Notice that f(0) =300e0.07(0) = 300e0 = 300, the original number of trout.)

In the year 2003, t = 3. g(3) = 300e0.07(3) = 300e0.21 = 300(1.2337) 370 trout in 2003.

In the year 2010, t = 10. g(10) = 300e0.07(10) = 300e0.70 = 300(2.0138) 604 trout in 2010.

Example:

Math130 ch09

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