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 LogarithmsLogarithms
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
The Laws of Logs
Module C2
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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.
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
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
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
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
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
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.
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 )
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 )
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
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
The Laws of Logs
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.
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.
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 )
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 )
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