1 C ollege A lgebra Inverse Functions ; Exponential and Logarithmic Functions (Chapter4) L:18 1...

1

College AlgebraInverse Functions ; Exponential

and Logarithmic Functions(Chapter4)

L:18

1

University of PalestineIT-College

2

Definition and Graph of the Natural Exponential Function

/ /١٤٤٤ ٠٩ ٢٩

The Natural Base eAn irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately,

The number e is called the natural base. The function f (x) = ex is called the natural exponential function.

2.71828...e

-1

f (x) = ex

f (x) = 2x

f (x) = 3x

(0, 1)

(1, 2)

1

2

3

4

(1, e)

(1, 3)

5

Solving Exponential Equations, where x is in the exponent, BUT the bases DO NOT MATCH.

Step 1: Isolate the exponential expression.

Get your exponential expression on one side everything outside of the exponential expression on the other side of your equation.

Step 2: Take the natural log of both sides.

The inverse operation of an exponential expression is a log. Make sure that you do the same thing to both sides of your equation to keep them equal to each other.

Step 3: Use the properties of logs to pull the x out of the exponent.

Step 4: Solve for x.

Now that the variable is out of the exponent, solve for the variable using inverse operations to complete the problem.

6

Example 1: Solve the exponential equation Round your answer to two decimal places.


This is already done for us in this problem.



7


8





9


10





11


12

Exponential Equations

•Solve 1

814

.

x

1

3

181

4

(4 ) 81

4 81

3

3

Def. of negative exponent

( )

4 4 Write 81 as a power of 4

Property (b)

Multiply b 1 y

mx

x

mn

x

n

x

a

x

a

x

13

Another Example

•Solve 3x + 1 = 27x 3

1 3

1 3

1 3 9

3

27

3 Write 27 as a power of 3.

(

3

3 ( )

3 3

1 3 9

2 10

5

)

Property (b)

Subtract 3 and 1.

Divide by 2.

x x

x x

x nx m mn

x x

x

x

a a

x

Objectives:After completing this tutorial, you should be able to:

1. Know the definition of a logarithmic function. 2. Write a log function as an exponential function and vice

versa. 3. Graph a log function. 4. Evaluate a log. 5. Be familiar with and use properties of logarithms in

various situations.6. Solve logarithmic equations.

sections 4.5,4.6,4.7& Equations Logarithmic Functions

Definition of Log Function

For all real numbers y, and all positive numbers a (a > 0) and x, where a 1:

Meaning of logax

A logarithm is an exponent; logax is the exponent to which the base a must be raised to obtain x.

(Note: Logarithms can be found for positive numbers only)

A LOG IS ANOTHER WAY TO WRITE AN EXPONENT.

log if and only if .yay x x a

Location of Base and Exponent in Exponential and Logarithmic Forms

Logarithmic form: y = logb x Exponential Form: by = x. Logarithmic form: y = logb x Exponential Form: by = x.

Exponent Exponent

Base Base

Example : Express the logarithmic equation exponentially

We want to use the definition that is above: if and only if .

Examples

Write each equation in its equivalent exponential form.a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y

Solution With the fact that y = logb x means by = x,

c. log3 7 = y or y = log3 7 means 3y = 7.

a. 2 = log5 x means 52 = x.

Logarithms are exponents.


b. 3 = logb 64 means b3 = 64.



Evaluating Logs

Step 1: Set the log equal to x.

Step 2: Use the definition of logs shown above to write the equation in exponential form.

Step 3: Find x.

Whenever you are finding a log, keep in mind that logs are another way to write exponents. You can always use the definition to help you evaluate.

Evaluating Logs

Example : Evaluate the expression without using a calculator.

Evaluatea. log2 16 b. log3 9 c. log25 5

Solution

log25 5 = 1/2 because 251/2 = 5.25 to what power is 5?c. log25 5

log3 9 = 2 because 32 = 9.3 to what power is 9?b. log3 9

log2 16 = 4 because 24 = 16.2 to what power is 16?a. log2 16

Logarithmic Expression Evaluated

Question Needed for Evaluation

Logarithmic Expression

Text Example

Graphing Log Functions

Characteristics of the Graph of f(x) = logax

The points (1, 0), and (a, 1) are on the graph.

If a > 1, then f is an increasing function; if 0 < a < 1, then f is a decreasing function.

The y-axis is a vertical asymptote.

The domain is (0, ), and the range is (, ).

1, 1 ,

a

Example

Graph Write

in exponential form

as

Now find some ordered pairs.

1/ 4( ) logf x x

1/ 4( ) logf x y x

1

4

y

x

14

21/16

01

yx

Graph Write

in exponential form

as

Now find some ordered pairs.

5( ) logf x x

10.2

15

01

yx

5yx

5( ) logf x x

Example

Translated Logarithmic Functions

Graph the function.

The vertical asymptote is x = 1.

To find some ordered pairs, use the equivalent exponent form.

3( ) log ( 1) f x x

3log ( 1)

1 3

3 1

y

y

y x

x

x

Translated Logarithmic Functions continued

Graph To find some ordered

pairs, use the equivalent exponent form.

4( ) (log ) 1 f x x

4

4

1

log 1

1 log

4 y

y x

y x

x

Properties of Logarithms, For x > 0, y > 0, a > 0, a 1, and any real number r:

The logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number.

Power Property

The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers.

Quotient Property

The logarithm of a product of two numbers is equal to the sum of the logarithms of the numbers

Product Property

DescriptionProperty

log log loga a axy x y

log log loga a a

xx y

y

log logra ax r x

Using the Properties of Logarithms

Rewrite each expression. Assume all variables represent positive real numbers with a 1 and b 1.

a)

b)

c)

6

12log

7

4log 11

2loga

abc

w

6 6 6log log12

12 77

log

4 4 41/ 2log 11 log (11) lo

1

2g 11

22

log log log log log

log log log 2log

a a a a a

a a a a

abca b c w

wa b c w


Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with a 1 and b 1.

a)

b)

3 3 3log ( 1) log log 5x x

3log 5logy yr t

3 3 3 3log log log log( 1)

1 55

( )x x

x x

3

3 5

5

log log log log

l

53

og

y y y y

y

r r

r

t

t

t


Expand as much as possible. Evaluate without a calculator where possible

Inverse Properties of Logarithms

For a > 0, a 1:

By the results of this theorem:

log and log .a x xaa x a x

7 1log 07 10 35log 5 3

Inverse Property I

Inverse Properties of Logarithms

For b > 0, b 1:

By the results of this theorem:

Inverse Property II

,

b logb x = x

Basic Logarithmic Properties Involving One

Logb b = 1

because 1 is the exponent to which b must be raised to obtain b. (b1 = b).

Logb 1 = 0

because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).

Properties of Common Logarithms

General Properties Common Logarithms

1. logb 1 = 0 1. log 1 = 0

2. logb b = 1 2. log 10 = 1

3. logb bx = x 3. log 10x = x4. b logb x = x 4. 10 log x = x

log b b = 1 log b 1 = 0 log 4 4 = 1

log 8 1 = 0 3 log 3 6 = 6 log 5 5 3 = 3

2 log 2 7 = 7

Examples of Logarithmic Properties

Natural Logarithms

Logarithms with a base of e are referred to a natural logarithms.

So if f(x) = ex , then f(x) = loge x = lnx Recall, e = 2.71828

Properties of Natural Logarithms

General NaturalProperties Logarithms

1. logb 1 = 0 1. ln 1 = 0

2. logb b = 1 2. ln e = 1

3. logb bx = 0 3. ln ex = x4. b logb x = x 4. e ln x = x

Examples

log e e = 1

log e 1 = 0

e log e 6 = 6

log e e 3 = 3

Change-of-Base Theorem

For any positive real numbers x, a, and b, where a 1 and b 1:

logax =lnx/ lna

loglog .

logb

ab

xx

a

Examples

a) log512 b) log2.4

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm.

5

lnlog

ln2.4849

1.60941.5440

25

121

2

loglog

log

.3979

.3010

2

1.321

..4

9

4

Solving Exponential or Logarithmic Equations

Solving Logarithmic Equations

Solve each equation. a) b)

3

33

27log 3

64

27

64

3

4

3

4

x

x

x

x

9

3 / 2

1/ 2 3

3

3log

2

9

(9 )

3

27

x

x

x

x

x

Example

Solve 8x = 15

The solution set is {1.3023}.

8 15

8 15

ln8 ln15

ln15

ln

ln

ln

81.3023

x

x

x

x

x

Example

Solve continued 2 1 35 .3 x x

2 1 3

2 1 3

5 .3

5 .3

(2 1)ln5 ( 3)ln.3

2 ln5 ln5 ln.3 3ln.3

2 ln5 ln.3 3ln.3 ln5

(2ln5 ln.3) 3l

ln ln

n.3 ln5

x x

x x

x x

x x

x x

x

3ln.3 ln5

2ln5 ln.3ln.027 ln5

ln 25 ln.3ln.135

25ln

.3.4528

x

x

x

x

Example

Solve 3

300xe3

3

3 3

3

300

ln ln300

ln300

300

1.7867

x

x

e

e

x

x

x

Solve log ( 2) log ( 1) log ( 2) b b bx x x

2

2

log ( 2) log ( 1)( 2)

( 2) 3 2

0 4

0 ( 4)

0 4 0

0 4

b bx x x

x x x

x x

x x

x or x

x x

The only valid solution is x = 4.

Example

Example

Solve 2

2

2

5log log

l

84

58

48( 4) 5

8 32 5

9

og (

2

5 ) log 4 3

3

(

7

)

x

x

x

x

xx x

x x

x

x

x x

Example

Solve continued

2

2

4ln ln(8 )

1

48

1(8 )( 1) 4

8

ln( 4) ln( 1) ln )

4

9

(

8 4

8

8

xx

x

xx

xx x x

x x x

x x

x

x

x

x x 20 8 12

0 ( 6)( 2)

6 0 2 0

6 2

x x

x x

x or x

x or x

The only valid solution is x = 2.

Notes, review

Properties of Logarithms

loga a 1 a MaMlog loga

ra r

log log loga a aMN M N

log log loga a aMN

M N

log loga aNN

1

log logar

aM r M

1

3log

2

2

x

xxa

Write the following expression as the sum and/or difference of logarithms. Express all powers as factors.

log log loga a a

x x

xx x x

2

22 23

13 1

log log loga a ax x x2 23 1

212

3 2 1log log loga a ax x x

Write the following expression as a single logarithm.

1log12log4

1log3 xxx aaa

log log loga a ax x x3 142 1 1

log log loga a ax x x3 4 2 1 1

log loga ax x x3 4 2 1 1

loga

x xx

3 4 2 11

Most calculators only evaluate logarithmic functions with base 10 or base e. To evaluate logs with other bases, we use the change of base formula.

loglogloga

b

b

MMa

loglog

Ma

lnln

Ma

Calculate log5 63

logloglog5 63

635

lnln

635

2 574.

Practices

Solve: log4 3 2 2x

log4 3 2 2x

3 2 42x

3 2 16x 3 18x x 6

Solve: log log6 63 2 1x x log log6 63 2 1x x

log6 3 2 1x x

log62 6 1x x

x x2 16 6 x x2 16 6 x x2 12 0

x x 4 3 0x x 4 3 or

Solution set: {x | x = 3}

Solve: 4 642x

4 642x

4 42 3x x 2 3

x 5

Solve: 9 3 10 02x x

9 3 10 02x x

3 3 3 10 02 2x x

3 9 3 10 02x x

3 10 3 1 0x x

3 10 0 3 1 0x x or

3 10 3 1x x or No Solution x 0

Solution set: x x 0

Solve: 7 53 2x x

7 53 2x x

ln ln7 53 2x x

x xln ln7 3 2 5 x xln ln ln7 3 5 2 5

x xln ln ln7 3 5 2 5 x ln ln ln7 3 5 2 5

x

2 5

7 3 5

ln

ln ln1117.

1 C ollege A lgebra Inverse Functions ; Exponential and Logarithmic Functions (Chapter4) L:18 1...

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Transcript of 1 C ollege A lgebra Inverse Functions ; Exponential and Logarithmic Functions (Chapter4) L:18 1...