1. 47: More Logarithms and Indices
“Teach A Level Maths” Vol. 1: AS Core Modules
47: More Logarithms and Indices
© Christine Crisp

2. Module C2
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3. We need to be able to change between index forms for numbers and log forms.
We use
We’ll also develop some more laws of logs.

4. e.g. Write the following in a form using logarithms:
(b)
Solution:
The index, 3, is the log of 64 and 4 is the base.
(b)
e.g. Write the following without using logarithms:
(a)
(b)
Solution: (a)
(b)

5. Exercises
1. Write the following in a form using logarithms:
(a)
(b)
2. Write the following without using logarithms:
(a)
(b)
Solution: 1(a)
(b)
2(a)
(b)

6. We are not solving an equation!
Simplifying Logs
Some logs can be simplified.
We are not solving an equation!
e.g. 1 Simplify
This log can be simplified because we can write 9 in index form using the base 3.
The base, 3, is now the same as the base of the log
So,
since a log is an index!
In general,

7. Simplifying Logs
e.g. 2. Simplify (a) (b)
Solution: (a)
(b)

8. Exercises
1. Simplify the following log expressions:
(a)
(b)
(c)
(d)
Solution (a)
(b)
(c)
(d)

9. 2 useful results
There are 2 special cases we can get directly from the definition of a log.
Let x = 0,
By the law of indices,
So,
for all values of the base

10. 2 useful results
There are 2 special cases we can get directly from the definition of a log.
Let b = a,
Then
x = 1

11. 2 useful results
There are 2 special cases we can get directly from the definition of a log.
Let b = a,
Then
x = 1
So,

12. SUMMARY
The Definition of a Logarithm
Three Laws of Logarithms

13. Exercises
1. Simplify the following:
(a)
(b)
(c)
(d)
(a) 1
Ans:
(b) 0
(c) 19
(d) b

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. e.g. Write the following in a form using logarithms:
(b)
e.g. Write the following without using logarithms:
Solution: (a)
Solution:

17. Simplifying Logs
e.g. 2. Simplify (a) (b)
(b)
Solution: (a)

18. There are 2 special cases we can get directly from the definition of a log.
Let x = 0,
2 useful results
So,
By the law of indices,
for all values of the base

19. Let b = a,
x = 1
Then
So,

20. Three Laws of Logarithms
The Definition of a Logarithm
SUMMARY