Detailed Notes: Logarithms | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


LOGARITHM 
Logarithms are a fundamental concept in mathematics that provide a way to express very large or 
very small numbers more concisely. They are essentially the inverse operation to exponentiation, 
allowing us to solve equations where the variable is in the exponent. 
The basic idea of a logarithm is to answer the question: "To what power must a given base be raised to 
produce a given number?" For example, if we have the equation 2^x = 8, we can express this as a 
logarithm: log2(8) = 3, because 2 raised to the power of 3 equals 8. 
Logarithms have numerous applications in various fields, including science, engineering, and finance. 
They are particularly useful for simplifying calculations involving multiplication and division, as they 
allow these operations to be converted into addition and subtraction. This property made logarithms 
invaluable in the pre-calculator era and continues to be important in many areas of advanced 
mathematics and data analysis. 
1. DEFINITION 
Every positive real number ?? can be expressed in exponential form as ?? ?? = ?? where 'a' is also a 
positive real number different than unity and is called the base and 'x' is called an exponent. 
We can write the relation ?? ?? = ?? in logarithmic form as ?? ?? ?? ?? ? ? ?? = ?? . Hence ?? ?? = ?? ? ??????
?? ? ? ?? = ?? . 
Hence, logarithm of a number to some base is the exponent by which the base must be raised in order 
to get that number. 
Limitations of logarithm: ??????
?? ? ?? is defined only when 
(i) ?? > 0 
(ii) ?? > 0 
(iii) ?? ? 1 
Note : 
(i) For a given value of ?? , ??????
?? ? ?? will give us a unique value. 
(ii) Logarithm of zero does not exist. 
(iii) Logarithm of negative reals are not defined in the system of real numbers. 
Problem 1 : If ??????
4
? ? ?? = 1 . 5, then find the value of ?? . 
Solution: ???????
4
? ? ?? = 1 . 5 ? ?? = 4
3 / 2
? ?? = 8 
Problem 2 : If ??????
5
? ?? = ?? and ??????
2
? ?? = ?? , then prove that 
?? 4
?? 4
100
= 100
2 ?? - 1
 
Solution : ???????
5
? ?? = ?? ? ?? = 5
?? 
??????
2
? ?? = ?? ? ?? = 2
?? 
Page 2


LOGARITHM 
Logarithms are a fundamental concept in mathematics that provide a way to express very large or 
very small numbers more concisely. They are essentially the inverse operation to exponentiation, 
allowing us to solve equations where the variable is in the exponent. 
The basic idea of a logarithm is to answer the question: "To what power must a given base be raised to 
produce a given number?" For example, if we have the equation 2^x = 8, we can express this as a 
logarithm: log2(8) = 3, because 2 raised to the power of 3 equals 8. 
Logarithms have numerous applications in various fields, including science, engineering, and finance. 
They are particularly useful for simplifying calculations involving multiplication and division, as they 
allow these operations to be converted into addition and subtraction. This property made logarithms 
invaluable in the pre-calculator era and continues to be important in many areas of advanced 
mathematics and data analysis. 
1. DEFINITION 
Every positive real number ?? can be expressed in exponential form as ?? ?? = ?? where 'a' is also a 
positive real number different than unity and is called the base and 'x' is called an exponent. 
We can write the relation ?? ?? = ?? in logarithmic form as ?? ?? ?? ?? ? ? ?? = ?? . Hence ?? ?? = ?? ? ??????
?? ? ? ?? = ?? . 
Hence, logarithm of a number to some base is the exponent by which the base must be raised in order 
to get that number. 
Limitations of logarithm: ??????
?? ? ?? is defined only when 
(i) ?? > 0 
(ii) ?? > 0 
(iii) ?? ? 1 
Note : 
(i) For a given value of ?? , ??????
?? ? ?? will give us a unique value. 
(ii) Logarithm of zero does not exist. 
(iii) Logarithm of negative reals are not defined in the system of real numbers. 
Problem 1 : If ??????
4
? ? ?? = 1 . 5, then find the value of ?? . 
Solution: ???????
4
? ? ?? = 1 . 5 ? ?? = 4
3 / 2
? ?? = 8 
Problem 2 : If ??????
5
? ?? = ?? and ??????
2
? ?? = ?? , then prove that 
?? 4
?? 4
100
= 100
2 ?? - 1
 
Solution : ???????
5
? ?? = ?? ? ?? = 5
?? 
??????
2
? ?? = ?? ? ?? = 2
?? 
?
?? 4
?? 4
100
=
5
4 ?? · 2
4 ?? 100
=
( 10 )
4 ?? 100
=
( 100 )
2 ?? 100
= 100
2 ?? - 1
 
Problem 3 : The value of ?? , satisfying ??????
?? ? [ 1 + ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } ] = 0 is - 
(A) 4 
(B) 3 
(C) 2 
(D) 1 
Solution : 
1 + ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } = ?? 0
= 1 ? ? ? ? ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } = 0 ?
? ? 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) = 1 ? ? ??????
?? ? ( 1 + ??????
?? ? ?? ) = 0 ? ? ? 1 + ??????
?? ? ?? = 1 ? ? ?? ?? ?? ?? ? ?? = 0 ? ? ? ?? = 1 ? # ( ?? ) ? 
Do yourself - 1 : 
(i) Express the following in logarithmic form : 
(a) 81 = 3
4
 
(b) 0 . 001 = 10
- 3
 
(c) 2 = 128
1 / 7
 
(ii) Express the following in exponential form: 
(a) ?? ?? ?? 2
? 32 = 5 
(b) ??????
v 2
? 4 = 4 
(c) ??????
10
? 0 . 01 = - 2 
(iii) If ?? ?? ?? 2 v 3
? 1728 = ?? , then find ?? . 
2. FUNDAMENTAL IDENTITIES 
Using the basic definition of logarithm we have 3 important deductions : 
(a) ?? ?? ?? ?? ? 1 = 0 
i.e. logarithm of unity to any base is zero. 
(b) ??????
?? ? ?? = 1 
i.e. logarithm of a number to the same base is 1 . 
(c) ?????? 1
? ?? ? ? ?? = - 1 = ??????
?? ?
1
? ?? 
i.e. logarithm of a number to the base as its reciprocal is -1 . 
Note : ?? = ( ?? )
?? ?? ?? ?? ? ?? e.g. 2
?? ?? ?? 2
? 7
= 7 
3. THE PRINCIPAL PROPERTIES OF 
LOGARITHMS 
If ?? , ?? are arbitrary positive numbers where ?? > 0 , ?? ? 1 and ?? is any real number, then- 
(a) ?? ?? ?? ?? ? ???? = ??????
?? ? ?? + ??????
?? ? ?? 
Page 3


LOGARITHM 
Logarithms are a fundamental concept in mathematics that provide a way to express very large or 
very small numbers more concisely. They are essentially the inverse operation to exponentiation, 
allowing us to solve equations where the variable is in the exponent. 
The basic idea of a logarithm is to answer the question: "To what power must a given base be raised to 
produce a given number?" For example, if we have the equation 2^x = 8, we can express this as a 
logarithm: log2(8) = 3, because 2 raised to the power of 3 equals 8. 
Logarithms have numerous applications in various fields, including science, engineering, and finance. 
They are particularly useful for simplifying calculations involving multiplication and division, as they 
allow these operations to be converted into addition and subtraction. This property made logarithms 
invaluable in the pre-calculator era and continues to be important in many areas of advanced 
mathematics and data analysis. 
1. DEFINITION 
Every positive real number ?? can be expressed in exponential form as ?? ?? = ?? where 'a' is also a 
positive real number different than unity and is called the base and 'x' is called an exponent. 
We can write the relation ?? ?? = ?? in logarithmic form as ?? ?? ?? ?? ? ? ?? = ?? . Hence ?? ?? = ?? ? ??????
?? ? ? ?? = ?? . 
Hence, logarithm of a number to some base is the exponent by which the base must be raised in order 
to get that number. 
Limitations of logarithm: ??????
?? ? ?? is defined only when 
(i) ?? > 0 
(ii) ?? > 0 
(iii) ?? ? 1 
Note : 
(i) For a given value of ?? , ??????
?? ? ?? will give us a unique value. 
(ii) Logarithm of zero does not exist. 
(iii) Logarithm of negative reals are not defined in the system of real numbers. 
Problem 1 : If ??????
4
? ? ?? = 1 . 5, then find the value of ?? . 
Solution: ???????
4
? ? ?? = 1 . 5 ? ?? = 4
3 / 2
? ?? = 8 
Problem 2 : If ??????
5
? ?? = ?? and ??????
2
? ?? = ?? , then prove that 
?? 4
?? 4
100
= 100
2 ?? - 1
 
Solution : ???????
5
? ?? = ?? ? ?? = 5
?? 
??????
2
? ?? = ?? ? ?? = 2
?? 
?
?? 4
?? 4
100
=
5
4 ?? · 2
4 ?? 100
=
( 10 )
4 ?? 100
=
( 100 )
2 ?? 100
= 100
2 ?? - 1
 
Problem 3 : The value of ?? , satisfying ??????
?? ? [ 1 + ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } ] = 0 is - 
(A) 4 
(B) 3 
(C) 2 
(D) 1 
Solution : 
1 + ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } = ?? 0
= 1 ? ? ? ? ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } = 0 ?
? ? 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) = 1 ? ? ??????
?? ? ( 1 + ??????
?? ? ?? ) = 0 ? ? ? 1 + ??????
?? ? ?? = 1 ? ? ?? ?? ?? ?? ? ?? = 0 ? ? ? ?? = 1 ? # ( ?? ) ? 
Do yourself - 1 : 
(i) Express the following in logarithmic form : 
(a) 81 = 3
4
 
(b) 0 . 001 = 10
- 3
 
(c) 2 = 128
1 / 7
 
(ii) Express the following in exponential form: 
(a) ?? ?? ?? 2
? 32 = 5 
(b) ??????
v 2
? 4 = 4 
(c) ??????
10
? 0 . 01 = - 2 
(iii) If ?? ?? ?? 2 v 3
? 1728 = ?? , then find ?? . 
2. FUNDAMENTAL IDENTITIES 
Using the basic definition of logarithm we have 3 important deductions : 
(a) ?? ?? ?? ?? ? 1 = 0 
i.e. logarithm of unity to any base is zero. 
(b) ??????
?? ? ?? = 1 
i.e. logarithm of a number to the same base is 1 . 
(c) ?????? 1
? ?? ? ? ?? = - 1 = ??????
?? ?
1
? ?? 
i.e. logarithm of a number to the base as its reciprocal is -1 . 
Note : ?? = ( ?? )
?? ?? ?? ?? ? ?? e.g. 2
?? ?? ?? 2
? 7
= 7 
3. THE PRINCIPAL PROPERTIES OF 
LOGARITHMS 
If ?? , ?? are arbitrary positive numbers where ?? > 0 , ?? ? 1 and ?? is any real number, then- 
(a) ?? ?? ?? ?? ? ???? = ??????
?? ? ?? + ??????
?? ? ?? 
(b) ??????
?? ?
?? ?? = ??????
?? ? ?? - ??????
?? ? ?? 
(c) ??????
?? ? ?? ?? = ?? ??????
?? ? ?? 
Problem 4 : Find the value of 2 ?????? ?
2
5
+ 3 ?????? ?
25
8
- ?????? ?
625
128
 
Solution : ? 2 ?????? ?
2
5
+ 3 ?????? ?
25
8
+ ?????? ?
128
625
 
= ?????? ?
2
2
5
2
+ ?????? ? (
5
2
2
3
)
3
+ ?????? ?
2
7
5
4
 
= ?????? ?
2
2
5
2
·
5
6
2
9
·
2
7
5
4
= ?? ?? ?? ? 1 = 0 
Problem 5 : If ??????
?? ? ?? - ??????
?? ? ?? = ?? , ??????
?? ? ?? - ??????
?? ? ?? = ?? & ?? ?? ?? ?? ? ?? - ??????
?? ? ?? = ?? , then find the value of 
(
?? ?? )
?? - ?? × (
?? ?? )
?? - ?? × (
?? ?? )
?? - ?? 
Solution : 
???????
?? ? ?? - ??????
?? ? ?? = ?? ? ??????
?? ?
?? ?? = ?? ?
?? ?? = ?? ?? ? ? ?? ?? ?? ?? ? ?? - ??????
?? ? ?? = ?? ? ??????
?? ?
?? ?? = ?? ?
?? ?? = ?? ?? ? ???????
?? ? ?? - ??????
?? ? ?? = ?? ? ??????
?? ?
?? ?? = ?? ?
?? ?? = ?? ?? ? ? ? ? ( ?? ?? )
?? - ?? × ( ?? ?? )
?? - ?? × ( ?? ?? )
?? - ?? ? ? ?
= ?? ?? ( ?? - ?? ) + ?? ( ?? - ?? ) + ?? ( ?? - ?? )
= ?? 0
= 1 ? 
Problem 6 : If ?? 2
+ ?? 2
= 23 ???? , then prove that ?????? ?
( ?? + ?? )
5
=
1
2
( ?????? ? ?? + ?????? ? ?? ) . 
Solution : 
? ?? 2
+ ?? 2
= ( ?? + ?? )
2
- 2 ???? = 23 ???? ? 
Using (i) 
L.H.S = ?????? ?
( ?? + ?? )
5
= ?????? ?
5 v ????
5
=
1
2
?????? ? ???? =
1
2
( ?????? ? ?? + ?????? ? ?? ) = R.H.S 
Problem 7 : If ??????
?? ? ?? = ?? and ??????
?? ? ?? 2
= ?? , then ??????
?? ? v ???? is equal to (where ?? , ?? , ?? ? ?? +
- { 1 } )- 
(A) 
1
?? +
1
?? 
(B) 
1
2 ?? +
1
?? 
(C) 
1
?? +
1
2 ?? 
(D) 
1
2 ?? +
1
2 ?? 
Solution : 
???????
?? ? ?? = ?? ? ?? ?? = ?? ? ?? = ?? 1 / ?? ? ? ?? ?? ?? ?? ?? ?? ???? ?? ?? ?? = ?? 2
? ?? = ?? 2 / ?? ? ? ?? ?? ?? , ??????
?? ? v ???? = ??????
?? ?
v
?? 1 / ?? ?? 2 / ?? = ??????
?? ? ?? (
1
?? +
2
?? ) ·
1
2
=
1
2 ?? +
1
?? ? 
Page 4


LOGARITHM 
Logarithms are a fundamental concept in mathematics that provide a way to express very large or 
very small numbers more concisely. They are essentially the inverse operation to exponentiation, 
allowing us to solve equations where the variable is in the exponent. 
The basic idea of a logarithm is to answer the question: "To what power must a given base be raised to 
produce a given number?" For example, if we have the equation 2^x = 8, we can express this as a 
logarithm: log2(8) = 3, because 2 raised to the power of 3 equals 8. 
Logarithms have numerous applications in various fields, including science, engineering, and finance. 
They are particularly useful for simplifying calculations involving multiplication and division, as they 
allow these operations to be converted into addition and subtraction. This property made logarithms 
invaluable in the pre-calculator era and continues to be important in many areas of advanced 
mathematics and data analysis. 
1. DEFINITION 
Every positive real number ?? can be expressed in exponential form as ?? ?? = ?? where 'a' is also a 
positive real number different than unity and is called the base and 'x' is called an exponent. 
We can write the relation ?? ?? = ?? in logarithmic form as ?? ?? ?? ?? ? ? ?? = ?? . Hence ?? ?? = ?? ? ??????
?? ? ? ?? = ?? . 
Hence, logarithm of a number to some base is the exponent by which the base must be raised in order 
to get that number. 
Limitations of logarithm: ??????
?? ? ?? is defined only when 
(i) ?? > 0 
(ii) ?? > 0 
(iii) ?? ? 1 
Note : 
(i) For a given value of ?? , ??????
?? ? ?? will give us a unique value. 
(ii) Logarithm of zero does not exist. 
(iii) Logarithm of negative reals are not defined in the system of real numbers. 
Problem 1 : If ??????
4
? ? ?? = 1 . 5, then find the value of ?? . 
Solution: ???????
4
? ? ?? = 1 . 5 ? ?? = 4
3 / 2
? ?? = 8 
Problem 2 : If ??????
5
? ?? = ?? and ??????
2
? ?? = ?? , then prove that 
?? 4
?? 4
100
= 100
2 ?? - 1
 
Solution : ???????
5
? ?? = ?? ? ?? = 5
?? 
??????
2
? ?? = ?? ? ?? = 2
?? 
?
?? 4
?? 4
100
=
5
4 ?? · 2
4 ?? 100
=
( 10 )
4 ?? 100
=
( 100 )
2 ?? 100
= 100
2 ?? - 1
 
Problem 3 : The value of ?? , satisfying ??????
?? ? [ 1 + ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } ] = 0 is - 
(A) 4 
(B) 3 
(C) 2 
(D) 1 
Solution : 
1 + ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } = ?? 0
= 1 ? ? ? ? ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } = 0 ?
? ? 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) = 1 ? ? ??????
?? ? ( 1 + ??????
?? ? ?? ) = 0 ? ? ? 1 + ??????
?? ? ?? = 1 ? ? ?? ?? ?? ?? ? ?? = 0 ? ? ? ?? = 1 ? # ( ?? ) ? 
Do yourself - 1 : 
(i) Express the following in logarithmic form : 
(a) 81 = 3
4
 
(b) 0 . 001 = 10
- 3
 
(c) 2 = 128
1 / 7
 
(ii) Express the following in exponential form: 
(a) ?? ?? ?? 2
? 32 = 5 
(b) ??????
v 2
? 4 = 4 
(c) ??????
10
? 0 . 01 = - 2 
(iii) If ?? ?? ?? 2 v 3
? 1728 = ?? , then find ?? . 
2. FUNDAMENTAL IDENTITIES 
Using the basic definition of logarithm we have 3 important deductions : 
(a) ?? ?? ?? ?? ? 1 = 0 
i.e. logarithm of unity to any base is zero. 
(b) ??????
?? ? ?? = 1 
i.e. logarithm of a number to the same base is 1 . 
(c) ?????? 1
? ?? ? ? ?? = - 1 = ??????
?? ?
1
? ?? 
i.e. logarithm of a number to the base as its reciprocal is -1 . 
Note : ?? = ( ?? )
?? ?? ?? ?? ? ?? e.g. 2
?? ?? ?? 2
? 7
= 7 
3. THE PRINCIPAL PROPERTIES OF 
LOGARITHMS 
If ?? , ?? are arbitrary positive numbers where ?? > 0 , ?? ? 1 and ?? is any real number, then- 
(a) ?? ?? ?? ?? ? ???? = ??????
?? ? ?? + ??????
?? ? ?? 
(b) ??????
?? ?
?? ?? = ??????
?? ? ?? - ??????
?? ? ?? 
(c) ??????
?? ? ?? ?? = ?? ??????
?? ? ?? 
Problem 4 : Find the value of 2 ?????? ?
2
5
+ 3 ?????? ?
25
8
- ?????? ?
625
128
 
Solution : ? 2 ?????? ?
2
5
+ 3 ?????? ?
25
8
+ ?????? ?
128
625
 
= ?????? ?
2
2
5
2
+ ?????? ? (
5
2
2
3
)
3
+ ?????? ?
2
7
5
4
 
= ?????? ?
2
2
5
2
·
5
6
2
9
·
2
7
5
4
= ?? ?? ?? ? 1 = 0 
Problem 5 : If ??????
?? ? ?? - ??????
?? ? ?? = ?? , ??????
?? ? ?? - ??????
?? ? ?? = ?? & ?? ?? ?? ?? ? ?? - ??????
?? ? ?? = ?? , then find the value of 
(
?? ?? )
?? - ?? × (
?? ?? )
?? - ?? × (
?? ?? )
?? - ?? 
Solution : 
???????
?? ? ?? - ??????
?? ? ?? = ?? ? ??????
?? ?
?? ?? = ?? ?
?? ?? = ?? ?? ? ? ?? ?? ?? ?? ? ?? - ??????
?? ? ?? = ?? ? ??????
?? ?
?? ?? = ?? ?
?? ?? = ?? ?? ? ???????
?? ? ?? - ??????
?? ? ?? = ?? ? ??????
?? ?
?? ?? = ?? ?
?? ?? = ?? ?? ? ? ? ? ( ?? ?? )
?? - ?? × ( ?? ?? )
?? - ?? × ( ?? ?? )
?? - ?? ? ? ?
= ?? ?? ( ?? - ?? ) + ?? ( ?? - ?? ) + ?? ( ?? - ?? )
= ?? 0
= 1 ? 
Problem 6 : If ?? 2
+ ?? 2
= 23 ???? , then prove that ?????? ?
( ?? + ?? )
5
=
1
2
( ?????? ? ?? + ?????? ? ?? ) . 
Solution : 
? ?? 2
+ ?? 2
= ( ?? + ?? )
2
- 2 ???? = 23 ???? ? 
Using (i) 
L.H.S = ?????? ?
( ?? + ?? )
5
= ?????? ?
5 v ????
5
=
1
2
?????? ? ???? =
1
2
( ?????? ? ?? + ?????? ? ?? ) = R.H.S 
Problem 7 : If ??????
?? ? ?? = ?? and ??????
?? ? ?? 2
= ?? , then ??????
?? ? v ???? is equal to (where ?? , ?? , ?? ? ?? +
- { 1 } )- 
(A) 
1
?? +
1
?? 
(B) 
1
2 ?? +
1
?? 
(C) 
1
?? +
1
2 ?? 
(D) 
1
2 ?? +
1
2 ?? 
Solution : 
???????
?? ? ?? = ?? ? ?? ?? = ?? ? ?? = ?? 1 / ?? ? ? ?? ?? ?? ?? ?? ?? ???? ?? ?? ?? = ?? 2
? ?? = ?? 2 / ?? ? ? ?? ?? ?? , ??????
?? ? v ???? = ??????
?? ?
v
?? 1 / ?? ?? 2 / ?? = ??????
?? ? ?? (
1
?? +
2
?? ) ·
1
2
=
1
2 ?? +
1
?? ? 
4. BASE CHANGING THEOREM 
Can be stated as "quotient of the logarithm of two numbers is independent of their common base." 
Symbolically, ??????
?? ? ?? =
?? ?? ?? ?? ? ?? ?? ?? ?? ?? ? ?? , where ?? > 0 , ?? ? 1 , ?? > 0 , ?? ? 1 
Note : 
(i) ??????
?? ? ?? · ??????
?? ? ?? =
?? ?? ?? ? ?? ?? ?? ?? ? ?? ·
?? ?? ?? ? ?? ?? ?? ?? ? ?? = 1; hence ??????
?? ? ?? =
1
?? ?? ?? ?? ? ?? . 
(ii) ?? ?? ?? ?? ?? ? ?? = ?? ?? ?? ?? ?? ? ?? 
(iii) Base power formula : ??????
?? ?? ? ? ?? =
1
?? ??????
?? ? ?? 
(iv) The base of the logarithm can be any positive number other than 1, but in normal practice, only 
two bases are popular, these are 10 and ?? ( = 2 . 718 approx). Logarithms of numbers to the base 10 are 
named as 'common logarithm' and the logarithms of numbers to the base ?? are called Natural or 
Napierian logarithm. We will consider ?????? ? ?? as ??????
?? ? ?? or ?? ?? ?? . 
(v) Conversion of base e to base 10& viceversa : 
??????
?? ? ?? =
??????
10
? ?? ?? ?? ?? 10
? ?? = 2 . 303 × ??????
10
? ?? ; ? ??????
10
? ?? =
??????
?? ? ?? ??????
?? ? 10
= ??????
10
? ?? × ??????
?? ? ?? = 0 . 434 ??????
?? ? ?? 
Problem 8: If a, b, ?? are distinct positive real numbers different from 1 such that ( ??????
?? ? ?? · ??????
?? ? ?? -
??????
?? ? ?? ) + ( ??????
?? ? ?? · ?? ?? ?? ?? ? ?? - ??????
?? ? ?? ) + ( ??????
?? ? ?? · ?? ?? ?? ?? ? ?? - ?? ?? ?? ?? ? ?? ) = 0, then abc is equal to - 
(A) 0 
(B) e 
(C) 1 
(D) none of these 
Solution: ? ( ??????
?? ? ?? ??????
?? ? ?? - 1 ) + ( ??????
?? ? ?? · ??????
?? ? ?? - 1 ) + ( ??????
?? ? ?? ??????
?? ? ?? - 1 ) = 0 
?
?????? ? ?? ?????? ? ?? ·
?????? ? ?? ?????? ? ?? +
?????? ? ?? ?????? ? ?? ·
?? ?? ?? ? ?? ?????? ? ?? +
?????? ? ?? ?????? ? ?? ·
?????? ? ?? ?????? ? ?? = 3 
? ( ?????? ? ?? )
3
+ ( ?????? ? ?? )
3
+ ( ?????? ? ?? )
3
= 3 ?????? ? ?? ?? ?? ?? ?? ?? ?? 
? ( ?? ?? ?? ?? + ?????? ? ?? + ?????? ? ?? ) = 0 ? [ ? If ?? 3
+ ?? 3
+ ?? 3
- 3 ?? ?? ?? = 0, then ?? + ?? + ?? = 0 if ?? ? ?? ? ?? ] 
? ?? ?? ?? ? ?? ?? ?? = ?? ?? ?? ? 1 ? ?? ?? ?? = 1 
Problem 9: Evaluate : 81
1 / ?? ?? ?? 5
? 3
+ 27
?? ?? ?? 9
? 36
+ 3
4 / ?? ?? ?? 7
? 9
 
Solution : 
?81
?? ?? ?? 3
? 5
+ 3
3 ?? ?? ?? 9
? 36
+ 3
4 ?? ?? ?? 9
? 7
? ? ? = 3
4 ?? ?? ?? 3
? 5
+ 3
?? ?? ?? 3
? ( 36 )
3 / 2
+ 3
?? ?? ?? 3
? 7
2
? ? ? = 625 + 216 + 49 = 890 ? 
 
Page 5


LOGARITHM 
Logarithms are a fundamental concept in mathematics that provide a way to express very large or 
very small numbers more concisely. They are essentially the inverse operation to exponentiation, 
allowing us to solve equations where the variable is in the exponent. 
The basic idea of a logarithm is to answer the question: "To what power must a given base be raised to 
produce a given number?" For example, if we have the equation 2^x = 8, we can express this as a 
logarithm: log2(8) = 3, because 2 raised to the power of 3 equals 8. 
Logarithms have numerous applications in various fields, including science, engineering, and finance. 
They are particularly useful for simplifying calculations involving multiplication and division, as they 
allow these operations to be converted into addition and subtraction. This property made logarithms 
invaluable in the pre-calculator era and continues to be important in many areas of advanced 
mathematics and data analysis. 
1. DEFINITION 
Every positive real number ?? can be expressed in exponential form as ?? ?? = ?? where 'a' is also a 
positive real number different than unity and is called the base and 'x' is called an exponent. 
We can write the relation ?? ?? = ?? in logarithmic form as ?? ?? ?? ?? ? ? ?? = ?? . Hence ?? ?? = ?? ? ??????
?? ? ? ?? = ?? . 
Hence, logarithm of a number to some base is the exponent by which the base must be raised in order 
to get that number. 
Limitations of logarithm: ??????
?? ? ?? is defined only when 
(i) ?? > 0 
(ii) ?? > 0 
(iii) ?? ? 1 
Note : 
(i) For a given value of ?? , ??????
?? ? ?? will give us a unique value. 
(ii) Logarithm of zero does not exist. 
(iii) Logarithm of negative reals are not defined in the system of real numbers. 
Problem 1 : If ??????
4
? ? ?? = 1 . 5, then find the value of ?? . 
Solution: ???????
4
? ? ?? = 1 . 5 ? ?? = 4
3 / 2
? ?? = 8 
Problem 2 : If ??????
5
? ?? = ?? and ??????
2
? ?? = ?? , then prove that 
?? 4
?? 4
100
= 100
2 ?? - 1
 
Solution : ???????
5
? ?? = ?? ? ?? = 5
?? 
??????
2
? ?? = ?? ? ?? = 2
?? 
?
?? 4
?? 4
100
=
5
4 ?? · 2
4 ?? 100
=
( 10 )
4 ?? 100
=
( 100 )
2 ?? 100
= 100
2 ?? - 1
 
Problem 3 : The value of ?? , satisfying ??????
?? ? [ 1 + ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } ] = 0 is - 
(A) 4 
(B) 3 
(C) 2 
(D) 1 
Solution : 
1 + ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } = ?? 0
= 1 ? ? ? ? ??????
?? ? { 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) } = 0 ?
? ? 1 + ??????
?? ? ( 1 + ??????
?? ? ?? ) = 1 ? ? ??????
?? ? ( 1 + ??????
?? ? ?? ) = 0 ? ? ? 1 + ??????
?? ? ?? = 1 ? ? ?? ?? ?? ?? ? ?? = 0 ? ? ? ?? = 1 ? # ( ?? ) ? 
Do yourself - 1 : 
(i) Express the following in logarithmic form : 
(a) 81 = 3
4
 
(b) 0 . 001 = 10
- 3
 
(c) 2 = 128
1 / 7
 
(ii) Express the following in exponential form: 
(a) ?? ?? ?? 2
? 32 = 5 
(b) ??????
v 2
? 4 = 4 
(c) ??????
10
? 0 . 01 = - 2 
(iii) If ?? ?? ?? 2 v 3
? 1728 = ?? , then find ?? . 
2. FUNDAMENTAL IDENTITIES 
Using the basic definition of logarithm we have 3 important deductions : 
(a) ?? ?? ?? ?? ? 1 = 0 
i.e. logarithm of unity to any base is zero. 
(b) ??????
?? ? ?? = 1 
i.e. logarithm of a number to the same base is 1 . 
(c) ?????? 1
? ?? ? ? ?? = - 1 = ??????
?? ?
1
? ?? 
i.e. logarithm of a number to the base as its reciprocal is -1 . 
Note : ?? = ( ?? )
?? ?? ?? ?? ? ?? e.g. 2
?? ?? ?? 2
? 7
= 7 
3. THE PRINCIPAL PROPERTIES OF 
LOGARITHMS 
If ?? , ?? are arbitrary positive numbers where ?? > 0 , ?? ? 1 and ?? is any real number, then- 
(a) ?? ?? ?? ?? ? ???? = ??????
?? ? ?? + ??????
?? ? ?? 
(b) ??????
?? ?
?? ?? = ??????
?? ? ?? - ??????
?? ? ?? 
(c) ??????
?? ? ?? ?? = ?? ??????
?? ? ?? 
Problem 4 : Find the value of 2 ?????? ?
2
5
+ 3 ?????? ?
25
8
- ?????? ?
625
128
 
Solution : ? 2 ?????? ?
2
5
+ 3 ?????? ?
25
8
+ ?????? ?
128
625
 
= ?????? ?
2
2
5
2
+ ?????? ? (
5
2
2
3
)
3
+ ?????? ?
2
7
5
4
 
= ?????? ?
2
2
5
2
·
5
6
2
9
·
2
7
5
4
= ?? ?? ?? ? 1 = 0 
Problem 5 : If ??????
?? ? ?? - ??????
?? ? ?? = ?? , ??????
?? ? ?? - ??????
?? ? ?? = ?? & ?? ?? ?? ?? ? ?? - ??????
?? ? ?? = ?? , then find the value of 
(
?? ?? )
?? - ?? × (
?? ?? )
?? - ?? × (
?? ?? )
?? - ?? 
Solution : 
???????
?? ? ?? - ??????
?? ? ?? = ?? ? ??????
?? ?
?? ?? = ?? ?
?? ?? = ?? ?? ? ? ?? ?? ?? ?? ? ?? - ??????
?? ? ?? = ?? ? ??????
?? ?
?? ?? = ?? ?
?? ?? = ?? ?? ? ???????
?? ? ?? - ??????
?? ? ?? = ?? ? ??????
?? ?
?? ?? = ?? ?
?? ?? = ?? ?? ? ? ? ? ( ?? ?? )
?? - ?? × ( ?? ?? )
?? - ?? × ( ?? ?? )
?? - ?? ? ? ?
= ?? ?? ( ?? - ?? ) + ?? ( ?? - ?? ) + ?? ( ?? - ?? )
= ?? 0
= 1 ? 
Problem 6 : If ?? 2
+ ?? 2
= 23 ???? , then prove that ?????? ?
( ?? + ?? )
5
=
1
2
( ?????? ? ?? + ?????? ? ?? ) . 
Solution : 
? ?? 2
+ ?? 2
= ( ?? + ?? )
2
- 2 ???? = 23 ???? ? 
Using (i) 
L.H.S = ?????? ?
( ?? + ?? )
5
= ?????? ?
5 v ????
5
=
1
2
?????? ? ???? =
1
2
( ?????? ? ?? + ?????? ? ?? ) = R.H.S 
Problem 7 : If ??????
?? ? ?? = ?? and ??????
?? ? ?? 2
= ?? , then ??????
?? ? v ???? is equal to (where ?? , ?? , ?? ? ?? +
- { 1 } )- 
(A) 
1
?? +
1
?? 
(B) 
1
2 ?? +
1
?? 
(C) 
1
?? +
1
2 ?? 
(D) 
1
2 ?? +
1
2 ?? 
Solution : 
???????
?? ? ?? = ?? ? ?? ?? = ?? ? ?? = ?? 1 / ?? ? ? ?? ?? ?? ?? ?? ?? ???? ?? ?? ?? = ?? 2
? ?? = ?? 2 / ?? ? ? ?? ?? ?? , ??????
?? ? v ???? = ??????
?? ?
v
?? 1 / ?? ?? 2 / ?? = ??????
?? ? ?? (
1
?? +
2
?? ) ·
1
2
=
1
2 ?? +
1
?? ? 
4. BASE CHANGING THEOREM 
Can be stated as "quotient of the logarithm of two numbers is independent of their common base." 
Symbolically, ??????
?? ? ?? =
?? ?? ?? ?? ? ?? ?? ?? ?? ?? ? ?? , where ?? > 0 , ?? ? 1 , ?? > 0 , ?? ? 1 
Note : 
(i) ??????
?? ? ?? · ??????
?? ? ?? =
?? ?? ?? ? ?? ?? ?? ?? ? ?? ·
?? ?? ?? ? ?? ?? ?? ?? ? ?? = 1; hence ??????
?? ? ?? =
1
?? ?? ?? ?? ? ?? . 
(ii) ?? ?? ?? ?? ?? ? ?? = ?? ?? ?? ?? ?? ? ?? 
(iii) Base power formula : ??????
?? ?? ? ? ?? =
1
?? ??????
?? ? ?? 
(iv) The base of the logarithm can be any positive number other than 1, but in normal practice, only 
two bases are popular, these are 10 and ?? ( = 2 . 718 approx). Logarithms of numbers to the base 10 are 
named as 'common logarithm' and the logarithms of numbers to the base ?? are called Natural or 
Napierian logarithm. We will consider ?????? ? ?? as ??????
?? ? ?? or ?? ?? ?? . 
(v) Conversion of base e to base 10& viceversa : 
??????
?? ? ?? =
??????
10
? ?? ?? ?? ?? 10
? ?? = 2 . 303 × ??????
10
? ?? ; ? ??????
10
? ?? =
??????
?? ? ?? ??????
?? ? 10
= ??????
10
? ?? × ??????
?? ? ?? = 0 . 434 ??????
?? ? ?? 
Problem 8: If a, b, ?? are distinct positive real numbers different from 1 such that ( ??????
?? ? ?? · ??????
?? ? ?? -
??????
?? ? ?? ) + ( ??????
?? ? ?? · ?? ?? ?? ?? ? ?? - ??????
?? ? ?? ) + ( ??????
?? ? ?? · ?? ?? ?? ?? ? ?? - ?? ?? ?? ?? ? ?? ) = 0, then abc is equal to - 
(A) 0 
(B) e 
(C) 1 
(D) none of these 
Solution: ? ( ??????
?? ? ?? ??????
?? ? ?? - 1 ) + ( ??????
?? ? ?? · ??????
?? ? ?? - 1 ) + ( ??????
?? ? ?? ??????
?? ? ?? - 1 ) = 0 
?
?????? ? ?? ?????? ? ?? ·
?????? ? ?? ?????? ? ?? +
?????? ? ?? ?????? ? ?? ·
?? ?? ?? ? ?? ?????? ? ?? +
?????? ? ?? ?????? ? ?? ·
?????? ? ?? ?????? ? ?? = 3 
? ( ?????? ? ?? )
3
+ ( ?????? ? ?? )
3
+ ( ?????? ? ?? )
3
= 3 ?????? ? ?? ?? ?? ?? ?? ?? ?? 
? ( ?? ?? ?? ?? + ?????? ? ?? + ?????? ? ?? ) = 0 ? [ ? If ?? 3
+ ?? 3
+ ?? 3
- 3 ?? ?? ?? = 0, then ?? + ?? + ?? = 0 if ?? ? ?? ? ?? ] 
? ?? ?? ?? ? ?? ?? ?? = ?? ?? ?? ? 1 ? ?? ?? ?? = 1 
Problem 9: Evaluate : 81
1 / ?? ?? ?? 5
? 3
+ 27
?? ?? ?? 9
? 36
+ 3
4 / ?? ?? ?? 7
? 9
 
Solution : 
?81
?? ?? ?? 3
? 5
+ 3
3 ?? ?? ?? 9
? 36
+ 3
4 ?? ?? ?? 9
? 7
? ? ? = 3
4 ?? ?? ?? 3
? 5
+ 3
?? ?? ?? 3
? ( 36 )
3 / 2
+ 3
?? ?? ?? 3
? 7
2
? ? ? = 625 + 216 + 49 = 890 ? 
 
5. POINTS TO REMEMBER 
(i) If base of logarithm is greater than 1 then logarithm of greater number is greater. i.e. ??????
2
? 8 =
3 , ??????
2
? 4 = 2 etc. and if base of logarithm is between 0 and 1 then logarithm of greater number is 
smaller. i.e. ??????
1 / 2
? 8 = - 3 , ??????
1 / 2
? 4 = - 2 etc. 
?? ?? ?? ?? ? ?? < ??????
?? ? ?? ? [ ?? < ?? ? ???? ? ?? > 1 ? ?? > ?? ? ???? ? 0 < ?? < 1 ? 
(ii) It must be noted that whenever the number and the base are on the same side of unity then 
logarithm of that number to that base is positive, however if the number and the base are located on 
different side of unity then logarithm of that number to that base is negative. 
e.g. ??????
10
? v 10
3
=
1
3
; ??????
v 7
? 49 = 4 ; ?????? 1
2
? (
1
8
) = 3 ; ??????
2
? (
1
32
) = - 5 ; ??????
10
? ( 0 . 001 ) = - 3 
(iii) ?? +
1
?? = 2 if ?? is positive real number and ?? +
1
?? = - 2 if ?? is negative real number 
(iv) ?? = 2 , ?? ? ?? 
v ?? ?? = ?? 1 / ?? ? ?? ?? h 
 root of 'a' ('a' is a non negative number) 
Some important values : ??????
10
? 2 ˜ 0 . 3010 ; ??????
10
? 3 ˜ 0 . 4771 ; ???? ? 2 ˜ 0 . 693 , ???? ? 10 ˜ 2 . 303 
6. CHARACTERISTIC AND MANTISSA 
For any given number ?? , logarithm can be expressed as ??????
?? ? ? ?? = Integer + Fraction 
The integer part is called characteristic and the fractional part is called mantissa. When the value of 
?????? ? ?? is given, then to find digits of ' ?? ' we use only the mantissa part. The characteristic is used only 
in determining the number of digits in the integral part (if ?? = 1 ) or the number of zeros after 
decimal & before first non-zero digit in the number (if 0 < ?? < 1 ). 
Note : 
(i) The mantissa part of logarithm of a number is always non-negative ( 0 = ?? < 1 ) 
(ii) If the characteristic of ??????
10
? ? ?? be ?? , then the number of digits in ?? is ( ?? + 1 ) 
(iii) If the characteristic of ??????
10
? ? ?? be ( - ?? ) , then there exist ( ?? - 1 ) zeros after decimal in ?? . 
7. ANTILOGARITHM 
The positive real number ' ?? ' is called the antilogarithm of a number ' ?? ' if ?? ?? ?? ? ?? = ?? 
Thus, ?????? ? ?? = ?? ? ?? = ?? ?? ?? ?? ?? ?? ?? ? ??