Logarithms: Definition, Types and Ways to Find Log

Who invented the logarithm?

The technique of logarithms was openly submitted by Scottish mathematician John Napier in 1614, in a book named Mirifici Logarithmorum Canon is Descriptio (Description of the Wonderful Rule of Logarithms)

His definition was given in terms of relative rates:

“The logarithme, therefore, of any sine is a number very nearly representing the line which increased equally in the same time while the line of the whole sine decreased proportionally into that sine, both movements being equivalent timed and the beginning equally shift.”

The formulation was swiftly and widely met with acclaim. The works of Bonaventura Cavalieri (Italy), Edmund Wingate (France), Xue Fengzuo (China), and Johannes Kepler’s Chilias logarithmorum (Germany) helped spread the concept further

Understanding of logarithm

To understand the logarithm, firstly we have to revise the concept of exponents.


Exponent or power is a number that shows how many times a number is going to be multiplied.



it shows us that 6 is going to multiply 3 times as:

6*6*6 = 216

Definition of Logarithm:

Logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if

 bx= n,

in which case one writes

x = logbn.


52 = 25 ,  2 = log5 25

Types of Logarithm

There are two types of logarithm depending upon the base that we are using.

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Common Logarithm:

The logarithm with the base 10 is common logarithm.


Log10 35, log10

Natural logarithm:

The logarithm with the base e (e= 2.718281828459) is natural logarithm.


Loge 8, loge 1.2566

Methods of finding logarithm
  1. Using Calculator


A scientific calculator.

scientific calculator


Step 1

Turn on the calculator

Step 2

Find the log key (for base 10) or ln key (for base e) on the keypad of your calculator and press it.

find logarithm

Step 3

Press the log key and then write the number on the calculator whose log you want to find.

find logarithm

Step 4

Press = key to see the answer.


2. Using logarithmic table

A logarithm of a number is divided into two parts. One is known as the characteristic and the other is the mantissa.

We will find the characteristic first. To evaluate it, follow the given instructions.

 Find the integer portion

Also called the “characteristic”.

  • By trial and error, find integeral value of p such thatap< n and ap+1> n
  • For number 32.5, as 101< 32 and 102> 32 so, characteristic is 1.
  • For base-10 logs. Just count the digits left from the decimal and subtract one. As in 32.5, there are two digits on the left of the decimal, now subtract 1 and we get the characteristic.
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Suppose we have a number =  232.92 . And we want to evaluate its common logarithm using the table.

For characteristic, 102< 232 and 103> 232, hence its characteristic is 2.

Now we are supposed to calculate the mantissa of the given number.


  1. Choose the correct table.

To find a loga(n), you will have a loga table. Almost all log tables are for base-10 logarithms, called “common logs” as mentioned earlier.

  1. Find the correct cell
  • Look for the cell value at the following intersections, ignoring all decimal places.
  • The row labeled with the first two digits of the number
  • Column header with the third digit of number


We have a number 232.92. we are supposed to look at the row labeled with 23 in the Colom labeled with 2. The corresponding number is 3655




  1. Use the smaller chart for precise numbers

Some tables have a smaller set of columns on the right side of the chart. Use these to adjust answer if the number has four or more significant digits:

  • Stay in the same row.
  • Find small column naming ‘mean difference’ header with the fourth digit of number.
  • Add this to the last value.
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Now we have to look at row 23 and column 9 of mean difference. The value we get is 17.

Now adding it as 3655+17 = 3672.



  1. Prefix a decimal point

The log table only tells you the portion of your answer after the decimal point. This is called the “mantissa.”


3672 is the portion of the logarithm of a number after a decimal point.

The value which is before the decimal point is characteristic, that we already found previously. Now, combining both of these we get the logarithm.
The logarithm of a number 232.92 is 2.3672