Online Learning Summit

LOGARITHMS

CONTENTS



Definition



Laws of Logarithms



Types of Logarithm



Problem Solving

DEFINITIO

If is any real number and N

is a positive integer, then the product of numbers is defined as

   

� ^�

Base

(real number)

Exponent (integer)

EXPONEN T

Where is the index or exponent or power and is the base.

 

2

⁵

, 3

⁻²

, �

⁶

, �

⁻⁷

��� . EXAMPL

 

E:

LOGARITHMS

,

   .

Definitio

n:

Logarithm is the inverse function to

exponentiation

�

^�

= �

   

⋯⋯ ( 1 )

 

� =���

_�

( � )

   

Examp le

^:

⋯⋯ ( 2 )

 

& are equivalent  

LAWS OF LOGARITHMS

Formul

a Example

���

  _�

( �� ) = ���

_�

( � ) + ���

_�

( � ) ���

  ₂

( 3 ⋅ 5 ) =���

₂

3 + ���

₂

5

���  _�

(

^��

)

^{=����(�)−����(� )}

^���

  ³

( ¹⁷ 24 )

^=���3

^{1 7}

⁻

^���

³

^{2 4}

���  _�

(

_�^�

)

=� ���_�(_�) ���  ₃

(

5⁷

)

=7���₃5

���

  _�

� =1 ���

  ₂

2= 1

TYPES OF LOGARITHM

Common Logarithms

 The system of logarithms whose base is 10 is called the common

logarithm system. When the base is omitted, it is understood that base 10 is to be used

 Thus,

  

Natural Logarithms:

 The system of logarithms whose base is the Eulerian constant is

called the natural logarithm system.

When we want to indicate the base of a logarithm is we write.

 Thus,

  

NOTE: Since so . Here the digit 1 before decimal point is called the

characteristic

and the digits . 5377 after decimal point is called the

mantissa

of the log.

 

PROBLEM & SOLUTION

1

4

  ²

= 16

Using log of base 4 we get

���  ₄ 4²=���₄ 16

��

 

,2 ���

₄

4 = ���

₄

16

��

 

, 2=���

₄

16

2

3⁻²=1 9

 

Using log of base 3 we get

���

  ₃

3

⁻²=���₃

( ¹ 9 )

��

  ,−

2 ���

₃

3= ���

₃

( ¹ 9 )

��

  ,−

2= ���

₃

( ¹ 9 )

3 8⁻

2

3= 1 4  

Using log of base 8 we get

���₈8⁻

2

3=���₈

(

¹4

)

 

��

,−

2

3 ���

₈

8

=���₈

( ¹ 4 )

 

��

,−

2

3

=���₈

( ¹ 4 )

 

Express each of the following exponential form in logarithmic form:

PROBLEM & SOLUTION CONT..

4

log

  ₅

25 =2

By the definition of log, we get

25

 

=5

²

5 6  

By the definition of log, we get

64

 

=2

⁶

6

log₁

4

1

16 =2

 

By the definition of log, we get

1

16 = ( ¹ 4 )

²

 

Express each of the following logarithmic form in exponential form:

PROBLEM & SOLUTION CONT..

7. Find the logarithms of 1728 to the base

  We have 1728

After factorization by prime number, we get

1728

 

= 2

⁶

. 3

³

��

 

, 2

⁶

. ( √ ^{3 )}

⁶

⁼¹⁷²⁸

��

 

, ( 2 √ ^{3 )}

⁶

⁼¹⁷²⁸

According to definition of log we get

6

 

¿ log

_2√3

1728

∴

 

log

_2√3

1728 =6

PROBLEM & SOLUTION CONT..

8. Find if

Given that,

  ^{ }

10 10

1log 11 4 7 log 2

2    x ^{4 42 4 1}



^{7 4 7}



x      2 1 

 



 

4 16 4 7 4 7 2

   



4 16 28 16 7 2

   

 ^{4 44 16 7}

2

  



4 2 11 4 7 2

  



2 11 4 7

   x 

2 11 4 7 , 2 11 4 7 x       

2 11 4 7

   x 

As 2  11 4 7  2 so x   2 11 4 7

PROBLEM & SOLUTION CONT..

9. Prove that

2 log

 

� + 2 log �

²

+ 2 log �

³

+ … + 2 log �

^�

L.H.S. =   =

¿

 

( 1 +2 + 3 + … + � ) 2 log �

¿ �(�+1)

2 .2 log �  

  =   =

∴

 

2 log � + 2 log �

²

+ 2 log �

³

+ …+ 2 log �

 ^{� =}

(Proved)

EXERCISE

3 5 2 a c b 3 5 2 a c b

10. Express the logarithm of 11. F from of equation

12. Solve

13. Solve the equation

14. Calculate the value of from