1. 46: Indices and Laws of Logarithms
“Teach A Level Maths” Vol. 1: AS Core Modules
46: Indices and Laws of Logarithms
© Christine Crisp

2. Unknown Indices
We have met the graph of and seen that it represents growth or decay.
Because of important practical applications of growth and decay, we need to be able to solve equations of the type
where a and b are constants.
Equations with unknown indices are solved using logarithms. We will see what a logarithm is and develop some rules that help us to solve equations.

3. e.g. How would you solve ?
Ans: If we notice that

4. e.g. How would you solve
then, (1) becomes
(1)
Ans: If we notice that
We can use the same method to solve or

5. log Suppose we want to solve
We need to write 75 as a power ( or index ) of 10.
Tip: It’s useful to notice that, since 75 lies between 10 and 100 ( or ), x lies between 1 and 2.
This index is called a logarithm ( or log ) and 10 is the base.
Our calculators give us the value of the logarithm of 75 with a base of 10.
The button is marked
log
The value is ( 3 d.p. ) so,

6. x is the logarithm of 4 with a base of 10
A logarithm is just an index.
To solve an equation where the index is unknown, we can use logarithms.
e.g. Solve the equation giving the answer correct to 3 significant figures.
x is the logarithm of 4 with a base of 10
We write
( 3 s.f. )
In general if
then
index
= log

7. Memory aid a to the power of x = b In general if then In general if

8. Exercise
Solve the following equations giving the answers correct to 2 d.p.
(a)
(b)
Solution:
(a)
( 2 d.p. )
(b)
( 2 d.p. )

9. In the exercise, we saw that
Generalizing this,
In general if
then
This relationship is also true changing from the log form to the index form,

10. The equation
When the base, a, is 10, we found the equation is easy to solve.
e.g. Solve the equation
Solution:
e.g. To solve
we could write
BUT there are no values for logs with base 2 on our calculators so we can’t find this as a simple number.
We need to develop some laws of logs to enable us to solve a variety of equations with unknown indices or logs

11. A law of logs for
( from the calculator )
e.g.
Also,
( from calculator )
And,
( from calculator )

12. A law of logs for
( from the calculator )
e.g.
Also,
( from calculator )
And,
( from calculator )

13. A law of logs for
( from the calculator )
e.g.
Also,
( from calculator )
And,
( from calculator )
We get

14. A law of logs for
The same reasoning holds for any base, a, so
( the “power to the front ” law of logs )

15. Solving
e.g.1 Solve
( Notice that 2 < x < 3 since )
Solution:
We “take” logs
We don’t actually take the logs anywhere: we put them in, but the process is always called taking logs!
Using the “power to the front” law, we can simplify the l.h.s.
We used logs with base 10 because the values are on the calculator. However, any base could be used. You could check the result using the “ln” button ( which uses a base you will meet in A2 ).

16. Solving
e.g.2 Solve the equation
Solution: We must change the equation into the form before we take logs.
Divide by 100:
Take logs:
Using the “power to the front” law:

17. SUMMARY
The Definition of a Logarithm
The “Power to the Front” law of logs:
Solving the equation
Divide by n
“Take” logs
Use the power to the front law
Rearrange to find x.

18. Exercises
1. Solve the following equations giving the answers correct to 2 d.p.
(a)
(b)
(a)
“Take” logs:
( 2 d.p. )
(b)
“Take” logs:
( 2 d.p. )

19. Exercises
2. Solve the equation giving the answer correct to 2 d.p.
Solution: Divide by 200:
Take logs:
Power to the front:
Rearrange:
( 2 d.p. )

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

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

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

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

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

25. 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.

26. 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.

27. 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 )

28. 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 )

29. 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

30. 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:

32. 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.

33. x is the logarithm of 4 with a base of 10
A logarithm is just an index.
To solve an equation where the index is unknown, we can use logarithms.
e.g. Solve the equation giving the answer correct to 3 significant figures.
x is the logarithm of 4 with a base of 10
We write
In general if
then
= log
index
( 3 s.f. )
( from the calculator )

34. Generalizing this,
This relationship is also true changing from the log form to the index form,
In the exercise we used logs with a base of 10 but the definition holds for any base, so
so,
Base

35. SUMMARY
The Definition of a Logarithm
Solving the equation
“Take” logs
The “Power to the Front” law of logs:
Use the power to the front law
Rearrange to find x.
Divide by n

36. Solving
e.g.1 Solve
Solution:
We “take” logs
We don’t actually take the logs anywhere: we put them in, but the process is always called taking logs!
We used logs with base 10 because the values are on the calculator. However, any base could be used. You could check the result using the “ln” button ( which uses a base you will meet in A2 ).
Using the “power to the front” law, we can simplify the l.h.s.

37. Solution: We must change the equation into the form before we take logs.
e.g.2 Solve the equation
Using the “power to the front” law:
Divide by 100:
Take logs: