Page 136 - Mathematics_Form_Two

Logarithms

13. Given log 2 = 0.30103, Solving for y gives,
10
log 3 = 0.47712, y = log x
a
10
log 5 = 0.69897, log b
a
10
b
log 7 = 0.845098. But y = log x .
10
FOR ONLINE READING ONLY
Evaluate the following Therefore: log x= log x
a
b
log b
Mathematics for Secondary Schools (c) log 50 Evaluate each of the following:
a
(a) log 5
7
Example 6�20
9
(b) log 8
(b) log 125
12
(a) log 32
2
5
(d) log 10
(d) log 16
(c) log 81
4
3
4
Change of base formula Solution: log 32
10
(a) log 32=
2 log 2
The change of base for logarithms gives a 10 5
way to convert a logarithm from one base = log 2
10
to another. This is useful when evaluating log 2
10
logarithmic expressions of any baseby = 5log 2 = 5
10
translating expressions into common log 2
10
logarithms. (b) log 125= log 125
10
5
log 5
10
The rule of change of base states that: = log 5 3
10
log x log 5
log x= log b 3log 5
a
10
b
a
10
It is proved as follows: = log 5 = 3
10
10
Let y = log x (i) (c) log 81= log 81
3
log 3
b
10
By definition of logarithms, equation (i) = log 3 4
10
becomes: log 3
10
x
y
b = (ii) = 4log 3 = 4
10
Taking the logarithm to base a on both log 3
10
sides of equation (ii) gives, (d) log 16= log 16
10
4
log 4
( )
log b= log x (iii) 10 2
y
a
a
= log 4
10
Using the power rule for logarithms, log 4
10
equation (iii) becomes: 2log 4
= 10 = 2
y log b= log x log 4
a
a
10
130
Student's Book Form Two
11/10/2024 20:13:02
MATHEMATIC F2 v5.indd 130
MATHEMATIC F2 v5.indd 130 11/10/2024 20:13:02