LOGARITHMS

LOGARITHMS

(Prepared by P. Sivadas Master)

The Scottish mathematician John Napier invented logarithms. He designed the method of logarithms to make mathematical calculations very easy. Logarithms were very widely used before calculators were invented.  

Definition:

Logarithm of a number to a given base is the power to which the base must be raised to equal the number.

If   N = b^x , then x is said to be the logarithm of N, to the base b. It is written as

Log _b N = x

(Read as log N to the base b equal to x)

Types of logarithms

Logarithms are of two types - Natural Logarithms and Common Logarithms. Logarithms with base ‘e’, where e = 2.71828, is known as Natural Logarithms. Logarithms with base 10 are known as Common Logarithms.

Natural logarithm of number = 2.303 x Common logarithm of that number

 ln 42.58 = 2.303x log 42.58

    or

 Log _e 42.58 = 2.303xLog ₁₀ 42.58

Common Logarithms

A logarithm has two parts. They are Characteristic and Mantissa. Characteristic is the whole number part of a logarithm. Characteristic may be positive or negative or zero. Mantissa is the decimal portion, which is always positive, of the common logarithm.

How to find the Characteristic of a Logarithm

E.g.                 

                  The characteristic of log 4871635.8 is     = 7-1= 6

                  The characteristic of log 487163.58 is     = 6-1= 5

                  The characteristic of log 48716.358 is     = 5-1= 4

                  The characteristic of log 48716.358 is     = 4-1= 3

                  The characteristic of log 487.16358 is     = 3-1= 2

                  The characteristic of log 48.716358 is     = 2-1= 1

                  The characteristic of log 4.8716358 is     = 1-1= 0

If there is no non-zero digit before the decimal point, the characteristic of such number will be one greater than the number of zeros immediately after the decimal point. Such characteristic will be negative; the negative sign of which should be written above the number representing the characteristic and is read as Bar.

E.g. 

Characteristic from scientific notation of numbers

If we write the numbers in scientific notation, characteristic can easily be calculated

Number	Number in scientific notation	Exponential Part	characteristic of Logarithm
6358.05	6358.05  x 103	103	3
635.805	635.805  x 102	102	2
63.5805	6.35805  x 101	101	1
6.35805	6.35805  x 100	100	0
.635805	635805   x 10-1	10-1	-1
.0635805	6.35805  x 10-2	10-2	-2
.00635805	6.35805  x 10-3	10-3	-3
.000635805	6.35805  x 10-4	10-4	-4
.0000635805	6.35805  x 10-5	10-3	-5
.00000635805	6.35805  x 10-6	10-6	-6

How to find the Mantissa of a Logarithm

To find the mantissa we should refer the logarithm tables.4 digit logarithm tables are now available. So to find the mantissa of logarithm of a number, take the first 2 digit of the given number in left column, third digit in the respective middle column – find the common cell, read the number in that cell. The mean difference for the 4^th digit is found from the mean difference columns and add to the number we have read from the cell before. The sum will be the mantissa

Find the characteristic and mantissa and then the logarithms of the following numbers:

1.)  879.5426

         c = 3-1=2

         m = .9442

         log 879.5426 = c . m = 2. 9442

         log 879.5426= 2. 9442

        

2.) 73.1045

         c = 2-1=1

         m = .8640

         log 73.1045= c . m = 1.8640

         log 879.5426= 1.8640

3. ) 0.3147

         c = -(0+1)=

         m = .4979

         log 0.3147 = c . m = . 4979

         log 0.3147 = . 4979

4. ) .0000138

        

         m = .1399

        

5. 0.004854

        

         m = .6861