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Logarithm

Logarithm
When ax  = N , then we say that x = logarithm of N to the base a and write it as x = logaN . In simple words, it represents the power to which a number must be raised. Let me expand on that by giving a simpler example.

If we are asked, what would be the result if ‘a’ is multiplied with itself ‘b’ times; then your answer would be x = a*a*a*…. b (times). This can also be written as a^b. This is also known as ‘a raised to the power of b’

If we are asked, which number multiplied with itself ‘b’ times, will result in a; then you are asked for the value of x such that x*x*x*… b(times) = a
=> x^b = a
=> x = a^(1/b)
This is also known as ‘bth root of a’

If swe are asked, how many times should you multiply ‘a’ with itself to get ‘b’, that is where the concept of logarithm comes into the picture. You are asked for the value of ‘x’ such that
a*a*a…. x (times) = b
=> x = Log a b
This is also known as ‘Log b to the base a’

Examples to illustrate this :
How many times should you multiply 2 with itself to get 8? Answer is Log 2 8 = 3
How many times should you multiple 5 with itself to get 625? Answer is Log 5 625 = 4

Another way to understand this would be:
If  ax  = N, then  x = logaN
Where N is a POSITIVE number , “a” is a positive number OTHER THAN 1.

Since the log of a number is a value,  it has 2 parts:
  1. Integral part known as Characteristic
  2. Decimal part known as Mantissa
For example, Log 27 = 3 Log 3 = 3*0.4771 = 1.4313
In this case, the characterstic is 1 and the mantissa is 0.4313

There are 2 types of logarithms that are commonly used on the basis of bases:
  1. Natural logarithm : base of the number is “e” .
  2. Common logarithm : Base of the number is 10 . When the base is not mentioned , it can be taken as 10.

Important Properties of Logarithms

Logarithm Concepts
The following can be derived from the above properties.
Logarithm Concepts - 1

Example:
  • 34 = 81 ,   log3 81 = 4.
  • log3 10 = ? ( given values of  log3 2 and  log3 5)
log3 10 =  log3 ( 2* 5) =  log3 2 +  log3 5.

SOME POINTS TO REMEMBER :
  1. Characteristic of a number greater than unity for a common base is positive and is 1 less than the number of digits in the integral part.For example : Characteristic of log 1000 = 3 which is 1 less than the number of digits in 1000.
  2. For a number between 0 and 1 , the characteristic is negative and its magnitude is 1 more than the number of zeros after the decimal point. For example : Characteristic of log 0.001 = -3.
  3. log( x – y ) ≠ logx – logy
  4. log( x + y ) ≠ logx + logy

The questions on logarithms are generally very direct , but can be increased in difficulty level by introducing the concept of number of digits.
Firstly, have a look at the log values of some numbers ( base 10 ). The values are always mentioned in the question , but it is still advisable to memorise the values of numbers till 10.

NumberValue
10
20.301
30.4771
40.602
50.698
60.778
70.845
80.903
90.954

Let us find the number of digits in 3 100.
x = 3 100
Log x = 100 Log 3
= 47.71
Number of digits here will be 47 + 1 = 48.

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