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  =   \bar 2

4 . Exponential form  –    3 -3 = 1/27

Logarithmic form – log 3 (1/27) = -3  =   \bar 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   \bar 1

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 = 0 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

Basic Concepts of logarithms |log properties | logarithm tutorial | Excercise - 1 log rules |properties of logarithms | logarithm rules practice |

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My self Sivaramakrishna Alluri. Thank you for watching my blog friend

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