# Basic Concepts of logarithms | Log properties | Logarithm without base

In this session we know about basic concepts of logarithms like definition of logarithm with examples, common logarithms, natural logarithms and properties of logarithms.

## Log Definition and Properties | logarithm tutorial | Exercise – 1

### What is Logarithm

Take a relation between x, p and y is x = py

In the above relation, we are unable to decide the value of “y” for a given value of “x” because the relation may be the relative change does not follow the criteria of ratio or difference.

For example if you wont to get x = 25 in x = 5y, What should be the value of y?

Here power to which 5 must be raised to get 25  (i. e 5 x 5 = 25  & y = 2 )

Now defined the relation between “y” and “x”.

Here in 25 = 5y

“y” is called logarithm of 25 to the base 5 and it can be shortly written as  log 5 25  = 2

### Definition of Logarithm

Let “p” be a positive real number, p ≠ 0 and py = x. Then “y” is called the logarithm of “x” to the base “p” and is written as

log p x  = y , and conversely  if log p y  = x then px = y

In logarithm equation of log p x  = y, we can say that x >0 and p >o

( Since Logarithm to a negative base is not taken)

y = log p is called the logarithm form and py = x is called the exponential form of the equation connecting p , y & x #### Some examples of exponential forms and their logarithmic forms

1 . Exponential form  –    a y = N

Logarithmic form – log a N = y

2 . Exponential form  –    2 4 = 16

Logarithmic form – log 2 16 = 4

3 . Exponential form  –    10 -2 = 0.01

Logarithmic form – log 10 0.01 = -2  = 4 . Exponential form  –    3 -3 = 1/27

Logarithmic form – log 3 (1/27) = -3  = #### Some Examples on logarithms

Example – 1 : Find how many 3s required to multiply  for getting 27?

Answer:  To get 27 , we have to multiply 3 of the 3s to get 27  ( i. e 3 x 3 x 3 = 27 )

So the logarithm is 3

Now we  have to  multiply  3 times of the number 3 to get 27 is 3″   i.e log3 (27) = 3 &

read it as “logarithm of 27 to the base 3 is 3”  or  “log base 3 of 27 is 3”

Example – 2 : Find the value of log 4 1024

Answer: Here we  have to  multiply  5 times of the number  5  to get 1024 ( i.e 4 x 4 x 4 x 4 x 4 = 1024 )

So “logarithm of 1024 to the base  4  is 5 ”  or  “log base 4 of 1024 is 5

i.e log 4 1024  = 5

Example – 3 : Find the value of log 81 (3)

Answer:  Let log 81 (3) = x

Exponential form  – 81 x = 3  ⇔  (3 4) x =   ( 3  ) 1

No According to exponential rules 4x = 1 x = 1/4

Example – 4 : Find the value of log 2/3 (8/27)

Answer:   Let  log 2/3 (8/27) = x

Exponential form  – (2/3) x = 8/27  ⇔  (2/3) x =   ( 2/3  ) 3

No According to exponential rules  x = 3

#### Characteristic and Mantissa

The logarithm of a number consists of two parts – One is integral part and another is decimal part.

The integral part of the logarithm of a number is called its characteristic and the decimal part is called mantissa

For example log 10 25  = 1.3979

Here, Characteristic = 1 & Mantissa = 0.3979

Note: Mantissa is always written as positive number.

##### Rule for write Mantissa and Characteristic:

To make the mantissa positive ( In case the value of the logarithm of a number is negative), subtract 1 from the integral part and add to the decimal part.

For example log 10 (0.5) = – 0.3010

Thus 0.3010 =  ( 0 + 0.3010 ) = 0 0.3010  = 0 1 +  (  1 0.3010) = 1 + 0.699

So here mantissa is 0.699

When the characteristic is negative , it is represented by putting a bar on the number.

Thus in the above example instead of -1, we write ##### For number of log base 10

If the number of digits in a number is ” n” then the characteristic of logarithm of the number is ( n- 1)

If characteristic of logarithm of a number is “ n” then the number of digits in the number is ( n+1)

Note:

1. The characteristic of common logarithms of any positive number less than 1 is negative.

2. The characteristic of common logarithms of any positive number greater than 1 is positive.

3. If the logarithm to any base ‘ a ‘ given the characteristic ‘n ‘, then we can say that the number of integers possible is  given byn+1 – a n

### Common Logarithms

Logarithm to the base “10” are called common logarithm. It is denoted as log 10 x simply denoted as log x

i. e  while a logarithm is written without a base than it mean base is really 10

### Natural Logarithm

Logarithm to the base ” e” are called natural logarithm. It is denoted as log e x

Here “e” is a constant, which is an irrational number with an infinite, non-terminating value of e = 2.718

### Properties of Logarithm

1log a 1 = 0 for a > 0 , a 1  ( i.e Log 1 to any base is Zero)

Proof : Let  log a 1  = x . Then according to logarithm definition

ax = 1 , so it is possible only if x = 0

Therefore ax = a 0   ⇔  x = 0.

Hence log a 1 = 0 for all a > 0 , a 1

2log a a = 1 for a > 0 , a 1  ( i.e Log of a number to the same base is 1)

Proof : Let  log a a  = x . Then according to logarithm definition

ax = a , so it is possible only if x = 1

Therefore ax = a 1   ⇔  x = 1.

Hence log a a = 1 for all a > 0 , a 1

Logarithm Tutorial | Exercise – 2

Please go through the below link for logarithm applications with examples and solutions

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