Download Logarithms

Unit 4: Mathematics
Aims
•
Introduce the laws of
Logarithms.
Objectives
• Identify the 4 laws of
Logarithms
• Use the laws of
Logarithms to calculate
given formulas.
1
The laws of logarithms
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
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.
• If we write down that 64 =
then the
equivalent statement using logarithms is
On the calculator we presslog (64)÷log (8) = 2
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
The laws of logarithms
• The logarithm of 1000 to
base 10 is 3 (remember
103 = 1000). This is
written: log101000 = 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.
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
1
100
0
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
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
• 101.799340549.... so
• log(63) = 1.799340549....
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
• 101.7993 × 101.6128 (to four
decimals) The
multiplication can then be
done by adding the
indices:
• 10(1.7993 + 1.6128) = 103.4121
The laws of logarithms
• Example 1 :
• If
= 2 then
Example 2 :
We have 25 = .
Then
= 2.
The laws of logarithms
Example 3 :
If
Then:
• Example 4 :
• If
= 4 then
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:
Properties of Logs
Example 3
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
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.
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
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.
The second law of logarithms
Suppose x =
or equivalently
Suppose we raise both sides of x =
m:
= n.
to the power
Using the rules of indices we can write this as
Thinking of the quantity
logarithmic form is
as a single term, the
The second law of logarithms
•
Suppose x =
= n.
or equivalently
•
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
The third law of logarithms
As before, suppose
and
with equivalent logarithmic forms
a
and
(2)
The third law of logarithms
using the rules of indices.
In logarithmic form
which from (2) can be written
The logarithm of 1
Recall that any number raised
to the power zero is 1:
The logarithmic form of this is
Example
=
3x5
=
15 or
log (15) ÷ log (10)= 1.176091259
Therefore we can write this as:
10
1.176091259