2.1 Exponentials and logarithms

Lecture Outline

Exponent
Exponential function
Graphs of exponential functions
Logarithm
Graphs of logarithmic functions
Laws of

Introduction

What is exponent?

What is the basic idea of exponentiation?

Introduction

What is exponent? Exponent is an index or power.

What is the

What is the basic idea of exponentiation?

Introduction

Repeated addition

Repeated multiplication

What is the basic idea of exponentiation?

Introduction

An exponential function has the form
where a is constant
a

2.1.1 Sketch the graph of Exponential function

Let us see some graphs

Exponent, index, power
(variable)

base

f (x) = ax

Why a ≠1?

An exponential function has the form
where a is constant
a

An exponential function has the form
where a is constant
a

An exponential function has the form
where a is constant
a

Example:
The graph of g(x) is a reflection of the graph of

Example:
The graph of g(x) is a reflection of the graph of

(0,A) is the y-intercept

g (x) = Af (x) = Aax

Recall from Lecture 1.5 Vertical scaling

Example:

f (x) = Aax

Example:

h (x) = Aax

Notice that (0, A)=(0, 3)
is the y-intercept

g (x) = Aax

(0,A) is the y-intercept

What about a?

The value

On the graph, if we move one unit to the right

What about a? Notice from the table that the value of

Exponential Decay

For exponential graphs, the independent variable often represents time and so

2.1.2 Write an expression in logarithmic form

Exponential form vs Logarithmic form

Logarithmic

Exponential form vs Logarithmic form

Logarithmic form

Exponential form

base

exponent

Examples

log101000 =

log416 =

log327 =

log55 =

log31 =

log4(1/16)

Examples

log101000 =3

log416 =2

log327 =3

log55 =1

log31 =0

log4(1/16)

The logarithm with base 10 is called the common logarithm and

What are the numbers that base of logarithm can be?

Base of

What are the numbers that base of logarithm can be?

Base of

What are the numbers that base of logarithm can be?

Base of

What are the numbers that base of logarithm can be?

Base of

What are the numbers that base of logarithm can be?

Base of

What are the numbers that base of logarithm can be?

Base of

What are the numbers that base of logarithm can be?

Base of

What are the numbers that base of logarithm can be?

Base of

What are the numbers that base of logarithm can be?

Base of

Since the functions
f(x)=ex and g(x)=lnx
are inverses of each other, the corresponding graphs are

Example:
Sketch graphs of f(x)=2x and g(x)=log2x

2.1.4 Sketch the graph Logarithmic function

Horizontal
asymptote
y=0

Vertical

(b, B and C are constants

2.1.5 Apply the laws of logs

Logarithm Identities
The following identities hold for

As a sample, let us verify that the first identity holds.
Let
logax=b and logay=c
from

Relationship with Exponential Functions
The following two identities demonstrate that the operations

2.1.6 Solve Exponential and Logarithmic equations

Example 1
Solve the following equations
a. 5–x

Example 1
Solve the following equations
a. 5–x = 125 b. 32x – 1

b. In logarithmic form, 32x – 1 = 6 becomes

2x

Solution (1):

Solution (2):

Change the base of a log

Change-of-Base Formula

Example 3

Your turn (Example 4)

Solve simultaneous equations, giving your answers as exact

Your turn (Example 4)

Solutions:

Solve simultaneous equations, giving your answers as exact

Your turn (Example 5)

Your turn (Example 5)

Solutions:

Your turn (Example 6)

Your turn (Example 6)

Solutions:

Learning outcomes

At the end of this lecture, you should be able

Formulas to memorize

Laws of Logarithms:

Preview activity: Modelling with Exponential and Logarithmic functions
Watch this video
https://www.youtube.com/watch?v=0BSaMH4hINY

Preview activity: Modelling with Exponential and Logarithmic functions

How do you think…
Which