www.le.ac.uk

Logarithms

Department of MathematicsUniversity of Leicester

Contents

Taking Logs

Introduction

What is a Logarithm?

Properties of Logarithms

Inverse of Log

Introduction

Logarithms are to do with raising numbers to different powers.

If you write an equation in terms of logarithms it’s like phrasing the equation in a different way.

Next

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

IntroductionWhy do we use logarithms?

Next

• Phrasing the equation in a different way sometimes makes it easier to solve.

• Logarithms have a different kind of scale - the difference between two numbers is a ratio rather than a subtraction. Some relationships are more easy to spot if we’re working with logarithms.

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

What is a logarithm?

Next

• A logarithm, or log, is a function.

• It is written as:

• This is read as “log to the base a of b”

• It means:

• eg. because the power that you have to raise 2 by to get 8 is 3.

balog

What power do you have to raise a by to

get b?38log2

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

2

What is a logarithm?Question

What is ?

3 4

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

5

What is a logarithm?Question

What is the value of a in this expression: ?4625log a

15 25

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

What is a logarithm?

We said that a Logarithm is a function.It looks like this:

It is the inverse of the exponential function.

Next

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

x

y

1

What is a logarithm?Bases

You can use any base you want in a logarithm.

If we don’t write a base on our logarithm then we assume it is to the base 10.

‘ln’ means ‘natural log’ or ‘log to the base e’. e is a constant number, like π, and we sometimes use this because it has patterns that are seen in nature.

Next

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

Inverse of Log

The inverse of is . ie. the inverse of ‘log to the base a’ is ‘a to the power’

In other words:

Next

balog ba

baba )(log ba ba log

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

Inverse of Log: Proofs

Next

This means “What power do I have to raise a by to get ab

? The answer is b.

)(log ba a

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

means “What power do I have to raise a by to get b?” If I then take a to the power of this number, I get b.

ba ba log balog

Inverse of Log

‘Taking Logs’

‘Taking logs’ of both sides means putting the log function round them.

Take logs of to get

To get back, we do the inverse of log, which is ‘a to the power’: .

This simplifies to , which is what we started with.

7310

Next

)73(log)10(log aa

)73(log)10(log aa aa

7310

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

• So, we can take logs, but how do we work with them once we’ve got them?

• There are 3 main properties of logarithms:

1.

2.

3.

Properties of Logarithms

)log(loglog abba

Next

abab loglog

b

aba logloglog

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

1. When you add 2 logs, you get the log of their product.

Proof:

Properties of Logarithms

)log(loglog abba

Next

abCab

bBb

aAa

C

B

A

10 so ,log

10 so ,log

10 so ,log

)log(loglog1010

1010

101010

)log()log(log

abba

abba

CBA

CBA

Using the laws of indices

Let: Then:

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

2.

Proof:

Properties of Logarithms

Next

b

aba logloglog

Using the laws of indices

b

aba

b

aba

CBA

CB

A

logloglog

1010

1010

1010

10

logloglog

b

aC

b

abBb

aAa

C

B

A

10 so ,log

10 so ,log

10 so ,log

Then: Let:

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

3.

Proof:

So .

Properties of Logarithms

Next

abab loglog

bb

a

abaabLHS a

cancel 10 base logandpower theto10 because ,

logloglog 10101010

b

a

aRHS ab

b

cancel 10 base logandpower theto10 because ,

log1010

RHSLHS

Note: this is , not . balog balog

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

• Remember the definition of log:

• For any positive number a:

• , because

• , because

• doesn’t exist,because is always > 0.

Properties of Logarithms

Next

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

01log a 10 a

1log aa aa 1

)0something(log a xa

bacb ca log

Inverse of Log

Example

Solve this:

Take logs:

Use property 3:

Rearrange for x:

Next

xx 311 45

)4log()5log( 311 xx

4log)31(5log)1( xx

4log35log

4log5log

x

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

Question

Write as a single logarithm.

4

127 log)log()log( wvu

4

127log wvu

4

1

27

log

w

vu

4

1

27

log

w

vu

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

Click here to see a hint

(Hide Hint)

Question

Solve

15

105log7log)3log( x

5 35

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

Click here to see the solution

(Hide Solution)

Question

Solve 515 103 x

53log

51 5

3log

51log 5

3

51log

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log

Click here to see the solution (Hide Solution)

Logarithms give us another way of writing equations.

We can take logs, take inverse logs, or use the definition.

We can use the three properties of logs to simplify equations.

Conclusion

Next

Taking Logs

IntroWhat is a

Logarithm?Properties of Logarithms

Inverse of Log