How To Find Out The Logarithm of a Given Number

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What is Logarithm?

Logarithm of a number with respect to a given base is the power to which the base must be raised to represent that number.

For example: 1000 = 10^3  where ^ indicates raise to the power.

Log1000 = 3

Four Standard Formulae of Logarithm –

  1. log(mn) = log(m) + log(n) – Product Formula
  2. log(m/n) = log(m) – log(n) – Quotient Formula
  3. log(mn) = nlog(m) – Power Formula    , where n in log(mn) is in superscript.
  4. loga(m) = logbm x logab – Base Change Formula   ,where a and b are in subscripts and not the lastmost b in logab.

Log Tables –

The logarithm of a number consists of two parts :

(i)                  Integral part, called characteristic.

(ii)                Decimal part, called mantissa.

(a)    Characteristic: It may be positive, negative or zero.  We use the following two rules for finding characteristic.

Rule 1 : If the given number > 1, its characteristic is positive.   Count the number of digits to the left of the decimal point.  Subtract 1 from this number.  This is the characteristic of the log of given number.

For example :

Characteristics of log of 1795.2 is 4 – 1 = 3

Characteristics of log of 16.73 is 2 – 1 = 1

Characteristics of log of 1.923 is 1 – 1 = 0

Rule 2 : If the given number is less than one, its characteristic is negative.  It is put under the bar.  Count the number of zeros immediately after the decimal point.  Add 1 to this number.  It gives us the characteristic of the log of the given number.

For example :

Characteristic of log of 0.7821 is – (0 + 1) = 1

Characteristic of log of 0.0618 is – (1 + 1) = 2

Characteristic of log of 0.000345 is – (3 + 1) = 4

(b)Mantissa : As mentioned above, it is the decimal part of log of a given number.  It is always positive, and can be read from the log tables as follows :

  1. In the given number, omit the decimal point and zeros in the beginning and at the end. For example, in the log of 0.001184, we have to look up the mantissa of 1184.
  2. Take first two digits i.e. 11 and locate it in the first vertical column of the four figure log table.
  3. Go through the horizontal row beginning with 11 and look up the value under the column headed by the third digit i.e. 8.  From the log tables, we find this number as 0719.
  4. Continue moving in the same horizontal row and note the number in the small differences column headed by the fourth digit i.e. 4.  This number is 15.
  5. Add 15 to 0719.  We get 0734.
  6. Therefore, mantissa = .0734.

Note : If the number consists of a single digit, put zeroes at the end of the number to complete four digits.  Read the mantissa as explained above.

For example : log7 = log7.000 = 0.8451

Mantissa for 7,70,700,0.7 etc. is the same.

This is all for now.

In my next article, I will show you how to find antilogarithm and some more examples to explain you in more better sense.

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