Introduction To Logarithms

Introduction to Logarithmic Functions

You were introduced to inverse functions.•Inverse functions is the set of ordered

pair obtained by interchanging the x and y values.

f(x)

f-1(x)

GRAPHS OF EXPONENTIALS AND ITS INVERSE

Introduction to Logarithmic Functions

•Inverse functions can be created graphically by a reflection on the y = x

axis.

y = x

f(x)

f-1(x)

GRAPHS OF EXPONENTIALS AND ITS INVERSE

Introduction to Logarithmic Functions

•A logarithmic function is the inverse of an exponential

function

•Exponential functions have the following characteristics:

Domain: {x є R}

Range: {y > 0}

Introduction to Logarithmic Functions

•Let us graph the exponential function y = 2x

•Table of values:

GRAPHS OF EXPONENTIALS AND ITS INVERSE

Introduction to Logarithmic Functions

•Let us find the inverse the exponential function

y = 2x•Table of values:

GRAPHS OF EXPONENTIALS AND ITS INVERSE

Introduction to Logarithmic Functions

•When we add the function f(x) = 2x to this graph, it is evident that the inverse is a reflection on the y = x axis

f-1(x)

f(x)

GRAPHS OF EXPONENTIALS AND ITS INVERSE

f(x) f-

1(x)

Introduction to Logarithmic Functions

•You will find the inverse of an exponential

algebraically•Write the process in your notes

FINDING THE INVERSE OF AN EXPONENTIAL

y = ax

Interchange x yx = ay

•We write these functions as:x = ay

y = logax

x = ay

base

exponent

y = logaaxx

baseexponent

Introduction to Logarithmic Functions

FINDING THE INVERSE OF AN EXPONENTIAL

y=axExponential

FunctionInverse of the Exponential

Function

x ylogLogarithmic Form

Introduction to Logarithmic Functions

CHANGING FORMSExample 1) Write the following into logarithmic form:a) 43 =

64

b) 45 = 256

c) 27 = 128

d) (1/3)x=27

4log 64 3

4log 256 5

2log 128 7

13

log 27 x

Introduction to Logarithmic Functions

CHANGING FORMSExample 2) Write the following into exponential form:

a) log264=6

b) log255=1/2

c) log81=0

d) log1/39=2

ANSWERS

Introduction to Logarithmic Functions

CHANGING FORMSExample 2) Write the following into exponential form:

a) log264=6

26 = 64

251/2 = 5

80 = 1

(1/3)2 = 1/9

b) log255=1/2c) log81=0

d) log1/39=2

Introduction to Logarithmic Functions

EVALUATING LOGARITHMSExample 3) Find the value of x for each example:

a) log1/327 = xb) log5x = 3

c) logx(1/9) = 2d) log3x = 0

ANSWERS

Introduction to Logarithmic Functions

EVALUATING LOGARITHMSExample 3) Find the value of x for each example:

a) log1/327 = x b) log5x = 3

c) logx(1/9) = 2

d) log3x = 0

(1/3)x = 27(1/3)x = (1/3)-3

x = -3

53 = xx = 125

x2 = (1/9)

x = 1/3

30 = xx = 1

Introduction to Logarithmic Functions

BASE 10 LOGSScientific calculators can perform logarithmic operations. Your calculator has

a LOG button.This button represents logarithms in BASE 10 or

log10Example 4) Use your calculator to find the value of each of the following:

a) log101000 b) log 50

c) log -1000=

3

= 1.69

9

= Out of Domain

What is a logarithm ?What is a logarithm ?

Solution: log2 8 3

We read this as: ”the log base 2 of 8 is equal

to 3”.

3Write 2 8 in logarithmic form.

Write 42 16 in logarithmic form.

Solution: log4 16 2

Read as: “the log base 4 of 16 is equal to 2”.

Solution:

Write 2 3 1

8 in logarithmic form.

log2

1

8 3

1Read as: "the log base 2 of is equal to -3".

8

Okay, so now it’s time for you to try some on

your own.

1. Write 72 49 in logarithmic form.

7Solution: log 49 2

log5 10Solution:

2. Write 50 1 in logarithmic form.

3. Write 10 2 1

100 in logarithmic form.

Solution: log10

1

100 2

Solution: log16 4 1

2

4. Finally, write 161

2 4

in logarithmic form.

Write log3 814 in exponential form

Solution: 34 81

Write log2

1

8 3 in exponential form.

Solution: 2 3 1

8

Okay, now you try these next three.

1. Write log10 100 2 in exponential form.

3. Write log27 3 1

3 in exponential form.

2. Write log5

1

125 3 in exponential form.

1. Write log10 100 2 in exponential form.

Solution: 102 100

2. Write log5

1

125 3 in exponential form.

Solution: 3 15

125

3. Write log27 3 1

3 in exponential form.

Solution: 271

3 3

Our final concern then is to determine why logarithms

like the one below are undefined.

Our final concern then is to determine why logarithms

like the one below are undefined.

Can anyone give us an

explanation ?

Can anyone give us an

explanation ?

2log ( 8)

One easy explanation is to simply rewrite this logarithm in exponential

form. We’ll then see why a negative value

is not permitted.First, we write the problem with a variable.

2y 8 Now take it out of the logarithmic form

and write it in exponential form.What power of 2 would gives us -8 ?23 8 and 2 3

1

8

Hence expressions of this type are undefined.

2log ( 8) undefined WHY?

2log ( 8) y