Lesson (7.3) Part I - Logarithms

8/8/2019 Lesson (7.3) Part I - Logarithms

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(7.3) IntroductionTo Logarithms


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Logarithms were originallydeveloped to simplify complex

arithmetic calculations.

They were designed to transform

multiplicative processes

into additive ones.

Logarithms were originallydeveloped to simplify complex

arithmetic calculations.

They were designed to transform

multiplicative processes

into additive ones.


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Today of course we have calculators

and scientific notation to deal with such

large numbers.

So at first glance, it would seem that

logarithms have become obsolete.

Today of course we have calculators

and scientific notation to deal with such

large numbers.

So at first glance, it would seem that

logarithms have become obsolete.


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Our first job is to

try to make somesense of

logarithms.

Our first job is to

try to make somesense of

logarithms.


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Our first question then

must be:

Our first question then

must be:

What is a logarithm ?What is a logarithm ?


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Of course logarithms have

a precise mathematicaldefinition just like all terms in

mathematics. So lets

start with that.


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The first, and perhaps the

most important step, inunderstanding logarithms is

to realize that they always

relate back to exponentialequations.


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You must be able to convert

an exponential equation into

logarithmic form and vice

versa.

So lets get a lot of practice with this !


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Definition ofLogarithm A logarithmic function is theinverse ofan

exponential function.

Definition: logb

y = x means bx = y

Example 1: log10100 = 2 means 102 = 100



Read as log base 10 of100 equals 2




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y = 5x

y = 2x

Exponentials with different bases


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MOREEXAMPLES


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Example 1:

Solution: log2 8 ! 3

We read this as: the log

base 2 of 8 is equal to 3.

3Write 2 8 in logarithmic form.!


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Example 1a:

Write 42! 16 in logarithmic form.

Solution: log4 16 ! 2

Read as: the logbase 4 of 16 is

equal to 2.


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It is also very important to be

able to start with a logarithmic

expression and change this

into exponential form.

This is simply the reverse of

what we just did.

It is also very important to be

able to start with a logarithmic

expression and change this

into exponential form.

This is simply the reverse of

what we just did.


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Example 1b:

Solution:

Write 2 3 1

8 in log arithmic form .

log21

8! 3

1ead as: "the log base 2 of is equal to -3".8


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Example 1:

Write log3 81 ! 4 in exp onential form

Solution: 34! 81


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Example 2:

Write log21

8! 3 in exp onential form .

Solution: 2 3 !1

8


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We now know that a logarithm is

perhaps best understood

as being

closely related to an

exponential equation.

In fact, whenever we get stuck

in the problems that follow

we will return to

this one simple insight.

We might even state a

simple rule.

We now know that a logarithm is

perhaps best understood

as being

closely related to an

exponential equation.

In fact, whenever we get stuck

in the problems that follow

we will return to

this one simple insight.

We might even state a

simple rule.


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When working with logarithms,if ever you get stuck, try

rewriting the problem in

exponential form.

When working with logarithms,if ever you get stuck, try

rewriting the problem in

exponential form.

Conversely, when working

with exponential expressions,

if ever you get stuck, try

rewriting the problemin logarithmic form.

Conversely, when working

with exponential expressions,

if ever you get stuck, try

rewriting the problemin logarithmic form.


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Lets see if this simple

rulecan help us solve some

of the following

problems.

Lets see if this simple

rulecan help us solve some

of the following

problems.


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Solution:

Lets rewrite the problem

in exponential form.

62 ! x

Were finished ! x = 36

6Solveforx: log 2x !

Example 1


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Solution:

5y!

1

25

Rewrite the problem in

exponential form.

Since 1

25! 5

2

5y

! 5 2

y! 2

5

1Solvefory: log

25y!

Example 2


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Example 3

Evaluate log 27.

Try setting this up like this:

Solution:

log 27 ! y Now rewrite in exponential form.

3y! 27

3y! 3

3

y ! 3


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These next two problems

tend to be some of the

trickiest to evaluate.

Actually, they are merely

identities and

the use of our simplerule

will show this.


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Example 4

Evaluate : log7 72

Solution:

Now take it out of the logarithmic form

and write it in exponential form.

log7 72 ! y First, we write the problem with a variable.

7y! 7

2

y ! 2


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Example 5

Eval at : 4l 4 16

Solution:4

l 4 16! y

First, we write the problem with a variable.

log4 y ! log4 16Now take it out of the exponential form

and write it in logarithmic form.

Just like 3

! 8 converts to l 8 ! 3y ! 16

Lesson (7.3) Part I - Logarithms

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