ICAI Notes- Logarithms | Quantitative Aptitude for CA Foundation PDF Download

 Page 1


1.23 RATIO AND PROPORTION, INDICES, LOGARITHMS
The logarithm of a number to a given base is the index or the power to which the base must be
raised to produce the number, i.e. to make it equal to the given number. If there are three quantities
indicated by say a, x and n, they are related as follows:
If a
x
 = n, where n > 0, a > 0 and a ? 1
then x is said to be the logarithm of the number n to the base ‘a’ symbolically it can be expressed
as follows:
After reading this unit a student will learn –
? After reading this unit, a student will get fundamental knowledge of logarithm and its
application for solving business problems.
Overview of Logarithms
Fundamental Laws
of Logarithms
Definition of
Logarithms
Logarithms of
the numbers
Quotient of two
numbers same 
Product of
two numbers
Anti
Logarithms
Change of
base
as base same base
 1.4 LOGARITHM:
© The Institute of Chartered Accountants of India
Page 2


1.23 RATIO AND PROPORTION, INDICES, LOGARITHMS
The logarithm of a number to a given base is the index or the power to which the base must be
raised to produce the number, i.e. to make it equal to the given number. If there are three quantities
indicated by say a, x and n, they are related as follows:
If a
x
 = n, where n > 0, a > 0 and a ? 1
then x is said to be the logarithm of the number n to the base ‘a’ symbolically it can be expressed
as follows:
After reading this unit a student will learn –
? After reading this unit, a student will get fundamental knowledge of logarithm and its
application for solving business problems.
Overview of Logarithms
Fundamental Laws
of Logarithms
Definition of
Logarithms
Logarithms of
the numbers
Quotient of two
numbers same 
Product of
two numbers
Anti
Logarithms
Change of
base
as base same base
 1.4 LOGARITHM:
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
1.24
log n = x
a
i.e. the logarithm of n to the base ‘a’ is x. We give some illustrations below:
(i) 2
4
 = 16 ? log 
2
16 = 4
i.e. the logarithm of 16 to the base 2 is equal to 4
(ii) 10
3
 = 1000 ? log
10
1000 = 3
i.e. the logarithm of 1000 to the base 10 is 3
(iii)
-3
1
5 =
125
 ? ?
? ?
? ?
? ?
5
1
log
125
 =  - 3
i.e. the logarithm of 
1
125
 to the base 5 is –3
(iv) 2
3
 = 8 ? log
2
8 = 3
i.e. the logarithm of 8 to the base 2 is 3
1. The two equations a
x 
= n and x = log
a
n are only transformations of each other and should be
remembered to change one form of the relation into the other.
2. The logarithm of 1 to any base is zero. This is because any number raised to the power zero
is one.
Since a
0
 = 1 , log
a
1 = 0
3. The logarithm of any quantity to the same base is unity. This is because any quantity raised
to the power 1 is that quantity only.
Since a
1
 = a , log
a
a = 1
 ILLUSTRATIONS:
1. If log
a 
1
2 =
6
, 
find the value of a.
We have a
1/6
 = 2 ? a = 
6
( 2 )
 
= 2
3
 = 8
2. Find the logarithm of 5832 to the base 3 ? 2.
Let us take 
3 2
log
5832 = x
We may write, ? ?
x 3 6 6 6 6
(3 2 ) =5832=8 729=2 3 =( 2 ) (3) =(3 2 )
Hence, x = 6
Logarithms of numbers to the base 10 are known as common logarithm.
1.4.1  Fundamental Laws of Logarithm
1. Logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers
to the same base, i.e.
log
a
mn = log
a
m + log
a
n
, where n > 0, a > 0 and a = 1
Remarks:
© The Institute of Chartered Accountants of India
Page 3


1.23 RATIO AND PROPORTION, INDICES, LOGARITHMS
The logarithm of a number to a given base is the index or the power to which the base must be
raised to produce the number, i.e. to make it equal to the given number. If there are three quantities
indicated by say a, x and n, they are related as follows:
If a
x
 = n, where n > 0, a > 0 and a ? 1
then x is said to be the logarithm of the number n to the base ‘a’ symbolically it can be expressed
as follows:
After reading this unit a student will learn –
? After reading this unit, a student will get fundamental knowledge of logarithm and its
application for solving business problems.
Overview of Logarithms
Fundamental Laws
of Logarithms
Definition of
Logarithms
Logarithms of
the numbers
Quotient of two
numbers same 
Product of
two numbers
Anti
Logarithms
Change of
base
as base same base
 1.4 LOGARITHM:
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
1.24
log n = x
a
i.e. the logarithm of n to the base ‘a’ is x. We give some illustrations below:
(i) 2
4
 = 16 ? log 
2
16 = 4
i.e. the logarithm of 16 to the base 2 is equal to 4
(ii) 10
3
 = 1000 ? log
10
1000 = 3
i.e. the logarithm of 1000 to the base 10 is 3
(iii)
-3
1
5 =
125
 ? ?
? ?
? ?
? ?
5
1
log
125
 =  - 3
i.e. the logarithm of 
1
125
 to the base 5 is –3
(iv) 2
3
 = 8 ? log
2
8 = 3
i.e. the logarithm of 8 to the base 2 is 3
1. The two equations a
x 
= n and x = log
a
n are only transformations of each other and should be
remembered to change one form of the relation into the other.
2. The logarithm of 1 to any base is zero. This is because any number raised to the power zero
is one.
Since a
0
 = 1 , log
a
1 = 0
3. The logarithm of any quantity to the same base is unity. This is because any quantity raised
to the power 1 is that quantity only.
Since a
1
 = a , log
a
a = 1
 ILLUSTRATIONS:
1. If log
a 
1
2 =
6
, 
find the value of a.
We have a
1/6
 = 2 ? a = 
6
( 2 )
 
= 2
3
 = 8
2. Find the logarithm of 5832 to the base 3 ? 2.
Let us take 
3 2
log
5832 = x
We may write, ? ?
x 3 6 6 6 6
(3 2 ) =5832=8 729=2 3 =( 2 ) (3) =(3 2 )
Hence, x = 6
Logarithms of numbers to the base 10 are known as common logarithm.
1.4.1  Fundamental Laws of Logarithm
1. Logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers
to the same base, i.e.
log
a
mn = log
a
m + log
a
n
, where n > 0, a > 0 and a = 1
Remarks:
© The Institute of Chartered Accountants of India
1.25 RATIO AND PROPORTION, INDICES, LOGARITHMS
Proof:
Let log
a
m = x so that a
x
 = m – (I)
Log
a
n = y so that a
y
 = n – (II)
Multiplying (I) and (II), we get
m × n = a
x
 × a
y
 = a
x+y
log
a
mn = x + y (by definition)
? log
a
mn = log
a
m + log
a
n
2. The logarithm of the quotient of two numbers is equal to the difference of their logarithms to
the same base, i.e.
log
a
m
n
= log
a
m
 
– log
a
n
Proof:
Let log
a
m = x so that a
x
 = m ————(I)
log
a
n = y so that a
y
 = n ———————(II)
Dividing (I) by (II) we get
x
x-y
y
m a
= =a
n a
Then by the definition of logarithm, we get
log
a
m
n
 
= x – y = log
a
m – log
a
n
Similarly,
a a a a
1
log = log 1 - log n = 0 - log n
n
= – log
a
n
 
[
?
 log
a
1
 
= 0]
Illustration I: log ½ = log 1 – log 2 = –log 2
3. Logarithm of the number raised to the power is equal to the index of the power multiplied
by the logarithm of the number to the same base i.e.
log
a
m
n
 = n log
a
m
Proof:
Let log
a
m = x so that a
x
 = m
Raising the power n on both sides we get
(a
x
)
n 
= (m)
n
a
xn 
= m
n
 
   (by definition)
log
a
m
n 
= nx
i.e. log
a
m
n 
= n log
a
m
© The Institute of Chartered Accountants of India
Page 4


1.23 RATIO AND PROPORTION, INDICES, LOGARITHMS
The logarithm of a number to a given base is the index or the power to which the base must be
raised to produce the number, i.e. to make it equal to the given number. If there are three quantities
indicated by say a, x and n, they are related as follows:
If a
x
 = n, where n > 0, a > 0 and a ? 1
then x is said to be the logarithm of the number n to the base ‘a’ symbolically it can be expressed
as follows:
After reading this unit a student will learn –
? After reading this unit, a student will get fundamental knowledge of logarithm and its
application for solving business problems.
Overview of Logarithms
Fundamental Laws
of Logarithms
Definition of
Logarithms
Logarithms of
the numbers
Quotient of two
numbers same 
Product of
two numbers
Anti
Logarithms
Change of
base
as base same base
 1.4 LOGARITHM:
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
1.24
log n = x
a
i.e. the logarithm of n to the base ‘a’ is x. We give some illustrations below:
(i) 2
4
 = 16 ? log 
2
16 = 4
i.e. the logarithm of 16 to the base 2 is equal to 4
(ii) 10
3
 = 1000 ? log
10
1000 = 3
i.e. the logarithm of 1000 to the base 10 is 3
(iii)
-3
1
5 =
125
 ? ?
? ?
? ?
? ?
5
1
log
125
 =  - 3
i.e. the logarithm of 
1
125
 to the base 5 is –3
(iv) 2
3
 = 8 ? log
2
8 = 3
i.e. the logarithm of 8 to the base 2 is 3
1. The two equations a
x 
= n and x = log
a
n are only transformations of each other and should be
remembered to change one form of the relation into the other.
2. The logarithm of 1 to any base is zero. This is because any number raised to the power zero
is one.
Since a
0
 = 1 , log
a
1 = 0
3. The logarithm of any quantity to the same base is unity. This is because any quantity raised
to the power 1 is that quantity only.
Since a
1
 = a , log
a
a = 1
 ILLUSTRATIONS:
1. If log
a 
1
2 =
6
, 
find the value of a.
We have a
1/6
 = 2 ? a = 
6
( 2 )
 
= 2
3
 = 8
2. Find the logarithm of 5832 to the base 3 ? 2.
Let us take 
3 2
log
5832 = x
We may write, ? ?
x 3 6 6 6 6
(3 2 ) =5832=8 729=2 3 =( 2 ) (3) =(3 2 )
Hence, x = 6
Logarithms of numbers to the base 10 are known as common logarithm.
1.4.1  Fundamental Laws of Logarithm
1. Logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers
to the same base, i.e.
log
a
mn = log
a
m + log
a
n
, where n > 0, a > 0 and a = 1
Remarks:
© The Institute of Chartered Accountants of India
1.25 RATIO AND PROPORTION, INDICES, LOGARITHMS
Proof:
Let log
a
m = x so that a
x
 = m – (I)
Log
a
n = y so that a
y
 = n – (II)
Multiplying (I) and (II), we get
m × n = a
x
 × a
y
 = a
x+y
log
a
mn = x + y (by definition)
? log
a
mn = log
a
m + log
a
n
2. The logarithm of the quotient of two numbers is equal to the difference of their logarithms to
the same base, i.e.
log
a
m
n
= log
a
m
 
– log
a
n
Proof:
Let log
a
m = x so that a
x
 = m ————(I)
log
a
n = y so that a
y
 = n ———————(II)
Dividing (I) by (II) we get
x
x-y
y
m a
= =a
n a
Then by the definition of logarithm, we get
log
a
m
n
 
= x – y = log
a
m – log
a
n
Similarly,
a a a a
1
log = log 1 - log n = 0 - log n
n
= – log
a
n
 
[
?
 log
a
1
 
= 0]
Illustration I: log ½ = log 1 – log 2 = –log 2
3. Logarithm of the number raised to the power is equal to the index of the power multiplied
by the logarithm of the number to the same base i.e.
log
a
m
n
 = n log
a
m
Proof:
Let log
a
m = x so that a
x
 = m
Raising the power n on both sides we get
(a
x
)
n 
= (m)
n
a
xn 
= m
n
 
   (by definition)
log
a
m
n 
= nx
i.e. log
a
m
n 
= n log
a
m
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
1.26
Illustration II: 1(a) Find the logarithm of 1728 to the base 2 ?3.
Solution: We have 1728 = 2
6
 × 3
3
 = 2
6
 × ( ?3)
6 
= (2 ?3)
6
; and so, we may write
    log 
2 ?3
 1728 = 6
1(b) Solve 
18
10 10 10
1
log 25 - 2log 3 + log
2
Solution: The given expression
25 18
2
10
1
2
10 10
= log - log 3 +log
18
10 10
10
= log 5 - log 9 +log
10
10 10
5x18
= log =log =1
9
1.4.2 Change of Base
If the logarithm of a number to any base is given, then the logarithm of the same number to any
other base can be determined from the following relation.
   
a b a
log m =log m log b ?? ?
a
b
a
log m
log m =
log b
Proof:
Let log
a
m = x, log
b
m
 
= y and log
a
b = z
Then by definition,
a
x 
= m, b
y
 = m and a
z
 = b
Also a
x 
= b
y
 = (a
z
)
y
 = a
yz
Therefore, x = yz
? log
a
m = log
b
m ? log
a
b
a
b
a
log m
log m =
log b
Putting m = a, we have
log
a
a = log
b
a ? log
a
b
? log
b
a
 
? log
a
b = 1, since log
a
a = 1.
Example 1: Change the base of log
5
31 into the common logarithmic base.
Solution: 
b
a
b
log x
Since log x =
log a
?
© The Institute of Chartered Accountants of India
Page 5


1.23 RATIO AND PROPORTION, INDICES, LOGARITHMS
The logarithm of a number to a given base is the index or the power to which the base must be
raised to produce the number, i.e. to make it equal to the given number. If there are three quantities
indicated by say a, x and n, they are related as follows:
If a
x
 = n, where n > 0, a > 0 and a ? 1
then x is said to be the logarithm of the number n to the base ‘a’ symbolically it can be expressed
as follows:
After reading this unit a student will learn –
? After reading this unit, a student will get fundamental knowledge of logarithm and its
application for solving business problems.
Overview of Logarithms
Fundamental Laws
of Logarithms
Definition of
Logarithms
Logarithms of
the numbers
Quotient of two
numbers same 
Product of
two numbers
Anti
Logarithms
Change of
base
as base same base
 1.4 LOGARITHM:
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
1.24
log n = x
a
i.e. the logarithm of n to the base ‘a’ is x. We give some illustrations below:
(i) 2
4
 = 16 ? log 
2
16 = 4
i.e. the logarithm of 16 to the base 2 is equal to 4
(ii) 10
3
 = 1000 ? log
10
1000 = 3
i.e. the logarithm of 1000 to the base 10 is 3
(iii)
-3
1
5 =
125
 ? ?
? ?
? ?
? ?
5
1
log
125
 =  - 3
i.e. the logarithm of 
1
125
 to the base 5 is –3
(iv) 2
3
 = 8 ? log
2
8 = 3
i.e. the logarithm of 8 to the base 2 is 3
1. The two equations a
x 
= n and x = log
a
n are only transformations of each other and should be
remembered to change one form of the relation into the other.
2. The logarithm of 1 to any base is zero. This is because any number raised to the power zero
is one.
Since a
0
 = 1 , log
a
1 = 0
3. The logarithm of any quantity to the same base is unity. This is because any quantity raised
to the power 1 is that quantity only.
Since a
1
 = a , log
a
a = 1
 ILLUSTRATIONS:
1. If log
a 
1
2 =
6
, 
find the value of a.
We have a
1/6
 = 2 ? a = 
6
( 2 )
 
= 2
3
 = 8
2. Find the logarithm of 5832 to the base 3 ? 2.
Let us take 
3 2
log
5832 = x
We may write, ? ?
x 3 6 6 6 6
(3 2 ) =5832=8 729=2 3 =( 2 ) (3) =(3 2 )
Hence, x = 6
Logarithms of numbers to the base 10 are known as common logarithm.
1.4.1  Fundamental Laws of Logarithm
1. Logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers
to the same base, i.e.
log
a
mn = log
a
m + log
a
n
, where n > 0, a > 0 and a = 1
Remarks:
© The Institute of Chartered Accountants of India
1.25 RATIO AND PROPORTION, INDICES, LOGARITHMS
Proof:
Let log
a
m = x so that a
x
 = m – (I)
Log
a
n = y so that a
y
 = n – (II)
Multiplying (I) and (II), we get
m × n = a
x
 × a
y
 = a
x+y
log
a
mn = x + y (by definition)
? log
a
mn = log
a
m + log
a
n
2. The logarithm of the quotient of two numbers is equal to the difference of their logarithms to
the same base, i.e.
log
a
m
n
= log
a
m
 
– log
a
n
Proof:
Let log
a
m = x so that a
x
 = m ————(I)
log
a
n = y so that a
y
 = n ———————(II)
Dividing (I) by (II) we get
x
x-y
y
m a
= =a
n a
Then by the definition of logarithm, we get
log
a
m
n
 
= x – y = log
a
m – log
a
n
Similarly,
a a a a
1
log = log 1 - log n = 0 - log n
n
= – log
a
n
 
[
?
 log
a
1
 
= 0]
Illustration I: log ½ = log 1 – log 2 = –log 2
3. Logarithm of the number raised to the power is equal to the index of the power multiplied
by the logarithm of the number to the same base i.e.
log
a
m
n
 = n log
a
m
Proof:
Let log
a
m = x so that a
x
 = m
Raising the power n on both sides we get
(a
x
)
n 
= (m)
n
a
xn 
= m
n
 
   (by definition)
log
a
m
n 
= nx
i.e. log
a
m
n 
= n log
a
m
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
1.26
Illustration II: 1(a) Find the logarithm of 1728 to the base 2 ?3.
Solution: We have 1728 = 2
6
 × 3
3
 = 2
6
 × ( ?3)
6 
= (2 ?3)
6
; and so, we may write
    log 
2 ?3
 1728 = 6
1(b) Solve 
18
10 10 10
1
log 25 - 2log 3 + log
2
Solution: The given expression
25 18
2
10
1
2
10 10
= log - log 3 +log
18
10 10
10
= log 5 - log 9 +log
10
10 10
5x18
= log =log =1
9
1.4.2 Change of Base
If the logarithm of a number to any base is given, then the logarithm of the same number to any
other base can be determined from the following relation.
   
a b a
log m =log m log b ?? ?
a
b
a
log m
log m =
log b
Proof:
Let log
a
m = x, log
b
m
 
= y and log
a
b = z
Then by definition,
a
x 
= m, b
y
 = m and a
z
 = b
Also a
x 
= b
y
 = (a
z
)
y
 = a
yz
Therefore, x = yz
? log
a
m = log
b
m ? log
a
b
a
b
a
log m
log m =
log b
Putting m = a, we have
log
a
a = log
b
a ? log
a
b
? log
b
a
 
? log
a
b = 1, since log
a
a = 1.
Example 1: Change the base of log
5
31 into the common logarithmic base.
Solution: 
b
a
b
log x
Since log x =
log a
?
© The Institute of Chartered Accountants of India
1.27 RATIO AND PROPORTION, INDICES, LOGARITHMS
10
5
10
log 31
log 31=
log 5
?
Example 2: 
3
10
9 4
log 8
Prove that = 3 log 2
log 16 log 10 
Solution: Change all the logarithms on L.H.S. to the base 10 by using the formula.
a
b
a
log
log
log b
x
x=
 
, we may write
3
? ? ?
3
10 10 10
3
10 10 10
log 8 log 2 3log 2
log 8
log 3 log 3 log   
? ? ?
4
10 10 10
9 2
10 10 10
log 16 log 2 4log 2
log 16
log 9 log 3 2log 3   
? ? ? ? ?
10
4 10 2
10 10 10
log 10 1 1
log 10 log 10 = 1
log 4 log 2 2 log 2   
? ?
? ? ? ?
10 10 10
10
10 10
3 log 2 2 log 3 2 log 2
L.H.S.= log 10 = 1
log 3 4log 2 1  
= 3 log
10
2 = R.H.S.
Logarithm Tables:
The logarithm of a number consists of two parts, the whole part or the integral part is called the
characteristic and the decimal part is called the mantissa where the former can be known by
mere inspection, the latter has to be obtained from the logarithm tables.
Characteristic:
The characteristic of the logarithm of any number greater than 1 is positive and is one less than
the number of digits to the left of the decimal point in the given number. The characteristic of the
logarithm of any number less than one (1) is negative and numerically one more than the number
of zeros to the right of the decimal point. If there is no zero then obviously it will
be –1. The following table will illustrate it.
Number Characteristic
3 7 1 One less than the number of digits to
4 6 2 3 3 the left of the decimal point
6.21 0
Number Characteristic
.8 –1 One more than the number of zeros on
.07 –2 the right immediately after the decimal point.
[ ]
[ ]
© The Institute of Chartered Accountants of India