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

2. Module C2
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3. Log laws for Multiplying and Dividing
We’ll develop the laws by writing an example with the numbers in index form.

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

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

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

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

8. Any positive integer could be used as a base instead of 10, so we get:
A similar rule holds for dividing.
If the base is missed out, you should assume it could be any base e.g might be base 10 or any other number.

9. SUMMARY
The Laws of Logarithms are:
1. Multiplication law
2. Division law
3. Power law
The definition of a logarithm:
leads to 4. 5. 6.

10. e.g. 1 Express the following in terms of
(a)
(b)
(c)
Solution:
(a)
( Law 1 )
(b)
( Law 3 )
(c) Either
( Law 2 )
( Law 4 ) Or
( Law 3 )

11. e.g. 2 Express in terms of and
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.
So,
( Law 1 )
( Law 3 )

12. e.g. 3 Express each of the following as a single logarithm in its simplest form:
(b)
Solution:
(a)
(b)
This could be simplified to

13. Exercise
1. Express the following in terms of
(a)
(b)
(c)
Ans:
(a)
(b)
(c)
2. Express in terms of and
Ans:
3. Express the following as a single logarithm in its simplest form:
(a)
(b)
(a)
(b)
Ans:

15. 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.

16. SUMMARY
The Laws of Logarithms are:
1. Multiplication law
2. Division law
3. Power law
The definition of a logarithm:
leads to 4. 5. 6.

17. (a)
( Law 1 )
(b)
( Law 3 )
(c) Either
( Law 2 )
( Law 4 )
Solution:
e.g. 1 Express the following in terms of
(c) Or

18. e.g. 2 Express in terms of and
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.
So,
( Law 1 )
( Law 3 )

19. (b)
e.g. 3 Express each of the following as a single logarithm in its simplest form:
(a)
Solution:
This could be simplified to