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5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function. For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic functions are the inverse of exponential functions Example: log 2 8 = 3 since 2 3 = 8 Read as: “log base 2 of 8”.

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Definition of a logarithmic function
Definition of a Logarithmic Function

Fory> 0 and b > 0, b ≠ 1,

logby = x if and only if bx=y

Note: Logarithmic functions are the inverse of exponential functions

Example: log28 = 3 since 23= 8

Read as: “log base 2 of 8”


Location of base and exponent in exponential and logarithmic forms

Exponent

Exponent

Base

Base

Location of Base and Exponent in Exponential and Logarithmic Forms

Logarithmic form:x= logby Exponential Form: bx=y


Basic logarithmic properties involving one
Basic Logarithmic Properties Involving One

logby = x if and only if bx = y

  • Logbb = __

    because 1 is the exponent to which b must be raised to obtain b.

    (b1 = b).

  • Logb 1 = __

    because 0 is the exponent to which b must be raised to obtain 1.

    (b0 = 1).

1

0


Popular bases have special names

Base 10

log 10x = log x is called a common logarithm

Base “e”

log e x = ln x is called the natural logarithm

or “natural log”

Popular Bases have special names


E and natural logarithm
e and Natural Logarithm

e is the natural base and is also called “Euler’s number”

: an irrational number (like ) and is approximately equal to 2.718281828...

Real Life Use: Compounding Interest problem

Remember the formula as n approaches +

The Natural logarithmof a number x(written as “ln (x)”) is the power to which e would have to be raised to equal x.

For example, ln(7.389...) is 2, because e2=7.389

Note: and ln(x) are inverse functions.


Inverse properties
Inverse properties

Since are inverse functions.

and

Proof:

Then

Proof:

Since and ln(x) are inverse functions.

and


Example 1

24

16

=

b.

log7 1

0

70

1

=

=

c.

log5 5

1

51

5

=

=

d.

log0.01

2

0.01

=

=

10 2

e.

log1/4 4

1

4

=

=

1

1

4

Rewrite in Exponential Form

logby = x is bx= y

EXPONENTIAL FORM

LOGARITHMIC FORM

a.

log2 16

4

=


Example 1

Rewrite in Exponential Form

EXPONENTIAL FORM

LOGARITHMIC FORM

log e x = ln x

=

f. ln

=

g. ln


Example 2

Rewrite in Logarithmic Form Form

logby = x is bx= y

EXPONENTIAL FORM

LOGARITHMIC FORM

a. =

b. =

c. =

d. =


Example 3

1

41/2

2

=

Guess, check, and revise.

log4 2

=

2

4?

64

=

What power of 4 gives 64?

43

64

=

Guess, check, and revise.

log4 64

3

=

4?

2

=

What power of 4 gives 2?

Evaluate Logarithmic Expressions

logby = x is bx= y

Evaluate the expression.

a.

log4 64

b.

log4 2


Example 3

?

1

9

=

What power of gives 9?

3

1

3

2

1

9

=

Guess, check, and revise.

3

log1/3 9

2

=

Evaluate Logarithmic Expressions

c.

log1/3 9

d.

Since


Example 4

Simplifying Exponential Functions

a.

Since

= 5

b.

Since

=


Example 4

Simplifying Exponential Functions

c.

Since

= 6

d.

Since

=


Homework ws 5 3 odd problems only
Homework WS 5.3 odd problems only


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