1. Unit 4: Mathematics Introduce the laws of Logarithms. Aims Objectives
Identify the 4 laws of Logarithms
Use the laws of Logarithms to calculate given formulas.

2. The laws of logarithms

3. The laws of logarithms Logarithms come in the form
We say this as "log of to base ".
But what does mean?
means "What power of 5 gives 25?“
The answer is 2 because , = 25 in other words
25 = 2

4. The laws of logarithms means "What power of 2 gives 16 ?“
The answer is 4 because , = 16 , in other words
So means "What power of a gives x ,?" Note that both a and x must be positive.

5. On the calculator we press- log (64)÷log (8) = 2
If we write down that 64 = then the equivalent statement using logarithms is
On the calculator we press-
log (64)÷log (8) = 2
a) 2 b) 5 c) 3 d) 4
e) 3 f) 1
2 g) 1
3 h) 1 4

6. The laws of logarithms
Indices can be applied to any base. The tables of logarithms most useful in computations use a base of 10. These are called Common Logarithms. Any base could be used in theory. Base 10 simplifies the work involved in calculations because our number system is base 10. We can apply the laws of indices as before to base 10.
103 × 104 = 1,000 × 10,000 = 10(3 + 4) = 107 = 10,000,000

7. The laws of logarithms
The logarithm of 1000 to base 10 is 3 (remember 103 = 1000). This is written: log = 34
Because base 10 is so important, it is assumed if no base is indicated. The above can also be written simply as
log (1000) = 3.

8. The laws of logarithms
Note that the indices 3 and 4 (above) tell us how many zeros the numbers 1,000 and 10,000 contain.
Here is a list of some whole number base 10 logarithms.
Number
Equivalent
Logarithm
10,000,000
107 7
1,000,000
106 6
100,000
105 5
10,000
104 4
1,000
103 3
100
102 2 10
101 1

9. The laws of logarithms
Note that the logarithm of 1 is 0. This is because 100 = 1. This makes sense. When you multiply a number by 1 you do not change its value. Correspondingly, if you add 0 to the index you leave it unchanged.
10 × 1 = 101 × 100 = 10(1 + 0) = 101 = 10

10. The laws of logarithms
There is more (much more) to logarithms than the whole number values discussed so far. A number like 63 will have as its logarithm a number between 1 and 2. In fact, 63 can be written as so
log(63) =

11. The laws of logarithms
Imagine that we wish to multiply two numbers, say, 63 and 41. By using tables of logarithms the two numbers can be written as
× (to four decimals) The multiplication can then be done by adding the indices:
10( ) =

12. The laws of logarithms Example 1 : If = 2 then Example 2 :
We have 25 = .
Then = 2.

13. 1. (a) x = 3to9
(b) 8 = 2x
(c) 27 = 3x
(d) x = 43
(e) y = 25
(f) y = 52
(a) 4 = log3 y
(b) x = log3 27
(c) 2 = log4m
(d) 5 = log3 y
(e) 5 = logx 32
(f) x = log4 64

14. The laws of logarithms
Example 3 : If
Then:
Example 4 :
If = 4 then

15. 3. (a) 81
(b) 3
(c) 10
(d) 64
(e) 243
(f) 8

16. Properties of Logs
Logs have some very useful properties which follow from their definition and the equivalence of the logarithmic form and exponential form. Some useful properties are as follows:

17. Properties of Logs

18. Example 3

19. Section 2
1. (a) log2 y/x
(b) log2 x
(c) log3 y - 1
(d) 2 log4(xy)
(e) 0

20. The Natural Logarithm and Exponential
The natural logarithm is often written as ln which you may have noticed on your calculator.
The symbol e symbolizes a special mathematical constant. It has importance in growth and decay problems. The logarithmic properties listed above hold for all bases of logs. If you see log x written (with no base), the natural log is implied. The number e can not be written

21. The Natural Logarithm and Exponential
exactly in decimal form, but it is approximately 2:718. Of course, all the properties of logs that we have written down also apply to the natural log. In particular,
are equivalent statements.
We also have = 1 and ln 1 = 0.

23. 1. (a) 0.34
(b) -0:14
(c) 0.89
(d) 1.86
(e) 44.70
(f) 1.62
(g) 9.03
(h) 0.10
(i) 3.41
(j) 4.65

26. The first law of logarithms
Suppose
then the equivalent logarithmic forms are
Using the first rule of indices
Now the logarithmic form of the statement

27. The first law of logarithms
But and
and so putting these results together we have
So, if we want to multiply two numbers together and find the logarithm of the result, we can
do this by adding together the logarithms of the two numbers. This is the first law.

28. The second law of logarithms
Suppose x = or equivalently = n.
Suppose we raise both sides of x = to the power m:
Using the rules of indices we can write this as
Thinking of the quantity as a single term, the logarithmic form is

29. The second law of logarithms
Suppose x = or equivalently = n.
Suppose we raise both sides of x = to the power m:
Using the rules of indices we can write this as
Thinking of the quantity as a single term, the logarithmic form is

31. The third law of logarithms
As before, suppose
and
with equivalent logarithmic forms a
and
(2)

32. The third law of logarithms
using the rules of indices.
In logarithmic form
which from (2) can be written

33. The logarithm of 1
Recall that any number raised to the power zero is 1:
The logarithmic form of this is

34. Example 10 = 3 x 5 = 15 or log (15) ÷ log (10)= 1.176091259
Therefore we can write this as: 10