Page 127 - Mathematics_Form_Two

Logarithms

15. A factory emits 5.6 10× 5 kg of carbon Example 6�6
dioxide annually. If the factory Express each of the following equations
reduces its emissions by 1.4 10× 4 kg according to the given instruction:
in a year, what is the new emission
level? (a) log 8 3= in exponential form.
2
1
3
−
(b) 5 = in logarithmic form.
FOR ONLINE READING ONLY
Concept of logarithms 125
When a number is expressed in power (c) 0.1 10= − 1 in logarithmic form.
form, it is written as a base raised to an Solution
x
exponent. For instance, if a = b , then ‘a’ (a) log 8 3= in exponential form is
2
is written in terms of base ‘b’ raised to an written as 2 = . Mathematics for Secondary Schools
8
3
exponent ‘x’ or ‘x’ is the logarithm of a to 1
3
−
base b. (b) 5 = in logarithmic form is
125
The exponent x is the number that shows − 3 log= 5   1   .
how many times a base is multiplied by  125 
itself to obtain a product. Thus, x is called (c) 0.1 10= − 1 in logarithmic form is
the logarithm of a to base b. Symbolically, − 1 log= ( )
0.1 .
x
this is written as log a = , where a > 0, 10
b
b 0 and b ≠ 1. This notation is called Example 6�7
logarithmic notation. For instance, 64
in exponential form can be expressed Solve for x in each of the following
as 64 = 2 . The exponent 6 is called the equations:
6
logarithm of 64 to base 2, written as (a) x = log 100
10
6 log 64.= 2 (b) 5 log−= (0.00001
)
Consider the following: (c) log x = 2 x
8
(i) 25 5= 2 is written as 2 log 25= 5 Solution
(ii) 1000 10= 3 is written as (a) Given x = log 100.
10
3 log 1000= 10 Writing in exponential form gives
x
2
(iii) 0.0001 10= − 4 is written as 100 = 10 . Thus, 10 = 10 x
−=
4 log 0.0001
10 Equating exponents gives x = 2.

3
3
64 = 4 is written as log 64 = . (b) Given –5 log 0.00001−= x .
4
64 8= 2 is written aslog 64 = . Writing in exponential form, gives
2
x =
5
−
8
0.00001.

In general, a b= x is written as x = log a But 0.00001 10= − 5 .
b
where a and b are positive real numbers and Thus, x = 10 .
5
5
−
−
x is a real number, b 0 and b ≠ 1. Therefore, x = 10.
2,
Note: When expressing the logarithm of (c) Given log x = .
8
a number, make sure to specify the base Writing in exponential form gives 8 = .
x
2
to which it refers. Therefore, x = 64.
121
Student's Book Form Two
11/10/2024 20:12:51
MATHEMATIC F2 v5.indd 121
MATHEMATIC F2 v5.indd 121 11/10/2024 20:12:51