48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

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48: More Laws of 48: More Laws of Logarithms Logarithms © Christine Crisp Teach A Level Maths” Teach A Level Maths” Vol. 1: AS Core Vol. 1: AS Core Modules Modules

Transcript of 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

Page 1: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

48: More Laws of 48: More Laws of LogarithmsLogarithms

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

Page 2: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

Module C2

"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

Page 3: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

Log laws for Multiplying and Dividing

We’ll develop the laws by writing an example with the numbers in index

form.

Page 4: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

A log is just an index, so to write this in index form we need the logs from the calculator.

2642 41516231 1010

10922642

0383 )2642(log10

4151623110 038310

42log10 26log10and 41516231

)1092(

So, )2642(log10

Page 5: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

A log is just an index, so to write this in index form we need the logs from the calculator. 42log10 26log10and 4151

2642 41516231 1010

10922642

0383 )2642(log10

4151623110 038310

So, )2642(log10

6231

)1092(

42log10

Page 6: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

A log is just an index, so to write this in index form we need the logs from the calculator. 42log10 26log10and 4151

2642 41516231 1010

10922642

0383 )2642(log10

4151623110 038310

So, )2642(log10 26log10

6231

)1092(

42log10

Page 7: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

A log is just an index, so to write this in index form we need the logs from the calculator. 42log10 26log10and 4151

2642 41516231 1010

yxxy 101010 loglog)(log In general,

10922642

0383 )2642(log10

4151623110 038310

So, )2642(log10 26log10

6231

)1092(

42log10

Page 8: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

Any positive integer could be used as a base instead of 10, so we get:

yxxy aaa loglog)(log

A similar rule holds for dividing.

yxy

xaaa logloglog

If the base is missed out, you should assume it could be any base e.g. might be base 10 or any other number.

2log

Page 9: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of LogsSUMMARY

yxxy aaa logloglog

yxy

xaaa logloglog

The Laws of Logarithms are:

xkx ak

a loglog

• 1. Multiplication law

• 2. Division law

• 3. Power law

The definition of a logarithm:

bxba ax log

01log aleads to 4. 1log aa5.

ka ka log6.

Page 10: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

15log(a) 53log 5log3log ( Law 1 )

(b) 16log 42log 2log4 ( Law 3 )

(c) Either

3

1log 3log1log ( Law 2 )

3log0 ( Law 4 )

3log

Solution:

e.g. 1 Express the following in terms of 5log3log,2log and

15log(a) (b) 16log (c)

3

1log

Or

3

1log 13log

3log ( Law 3 )

Page 11: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

e.g. 2 Express in terms of and )log( 2ba blogalog

2loglog ba

ba log2log

Solution: We can’t use the power to the front law

directly!( Why not? )

There is no bracket round the ab, so the square ONLY refers to the b.

)log( 2baSo, ( Law 1 )( Law 3 )

Page 12: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

3

25log

3

10log

125log4log2 1021

10 (b)

10log25log4log 10102

1021

21

25

104log

2

10

5

1016log10

2

32log10

e.g. 3 Express each of the following as a single logarithm in its simplest form:

3log2log5log (a) 125log4log2 1021

10 (b)

Solution:

(a) 3log2log5log

510 2log 2log5 10This could be simplified

to

1

Page 13: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

Exercise1. Express the following in terms of 5log3log,2log and

25log(a) (b) (c) 10

1log

3. Express the following as a single logarithm in its simplest form:

5log2log3log (a) 116log2log3 1021

10 (b)

Ans:

(a) 5log2 3log2log (b)

6log

(c) 5log2log

Ans:2

15log(a) (b)

5

16log10

2. Express in terms of andalog blogba 2log

Ans:

ba loglog2

Page 14: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

Page 15: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Page 16: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

SUMMARY

yxxy aaa logloglog

yxy

xaaa logloglog

The Laws of Logarithms are:

xkx ak

a loglog

• 1. Multiplication law

• 2. Division law

• 3. Power law

The definition of a logarithm:

bxba ax log

01log aleads to 4. 1log aa5.

ka ka log6.

Page 17: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

15log(a) 53log 5log3log ( Law 1 )

(b) 16log 42log 2log4 ( Law 3 )

(c) Either

3

1log 3log1log ( Law 2 )

3log0 ( Law 4 )

3log

Solution:

e.g. 1 Express the following in terms of 5log3log,2log and

15log(a) (b) 16log (c)

3

1log

Or

3

1log 13log

3log ( Law 3 )

Page 18: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

e.g. 2 Express in terms of and )log( 2ba blogalog

2loglog ba

ba log2log

Solution: We can’t use the power to the front law

directly!( Why not? )

There is no bracket round the ab, so the square ONLY refers to the b.

)log( 2baSo, ( Law 1 )( Law 3 )

Page 19: 48: More Laws of Logarithms © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

The Laws of Logs

3

25log

3

10log

125log4log2 1021

10 (b)

10log25log4log 10102

1021

21

25

104log

2

10

5

1016log10

2

32log10

e.g. 3 Express each of the following as a single logarithm in its simplest form:

3log2log5log (a) 125log4log2 1021

10 (b)

Solution:

(a) 3log2log5log

510 2log 2log5 10This could be simplified

to

1