Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.29

Now log 0.0031 lies between – 3 and – 2.

∴ log 0.0031 = – 3 + a positive proper fraction.

Hence the characteristic of such number is – 3.

Thus we arrive at the following rule :

Rule 2 : The characteristic of the logarithm of a decimal fraction is negative and is greater by one than the

number of zeroes between the decimal point and the first significant digit.

Thus the characteristic of log 0.234, log 0.0234, log 0.00234 are respectively – 1, – 2, – 3

Finding the Mantissa :

Let N be any number, the

N 10 or N 10 ,× p ÷ q

where p and q are positive integers evidently a number having

the same significant digit N.

Now, log (N 10× p) = log N + log 10p = log N + p log 10 = log N + p.

log (N 10÷ q) = log N – log 10q = log N – q log 10 = log N – q.

Thus we see that p is added to and q is subtracted from the characteristic of N, while the Mantissa remains

unaffected, in both cases.

Hence we get the following rule :

Rule 3 : The mantissa is the same for logarithm of all numbers which have the same significant digits (i.e., the

mantissa does not depend on the position of the decimal point).

for example : Let us consider the logarithms of the numbers 234500, 23.45, 0.02345, having given

log 2345 = 3.3701.

log 234500 = log (2345 × 100) = log 2345 + log 100 = 3.3701 + 2 = 5.3701.

log 23.45= log^2345100 =log2345 log100− = 3.3701 – 2 = 1.3701

log 0.02345 = log 1000002345 =log 2345 log100000−

= 2.3701, where 2 (read as two bar)

denotes that it is equivalent to – 2, while 0.3701 is + ve.

Thus, we see that he mantissa in every case is same.

Note : The characteristic of the logarithm of any number may be + ve or –ve, but is mantissa is always + ve.

USE OF LOGARITHMIC TABLE :

It must be observed that only approximate values can be obtained from the table, correct upto 4 decimal

places. The main body of the table gives the mantissa of the logarithm of numbers of 3 digits or less, while

the mean difference table provides the increment for the fourth digit.

Let us find the logarithm of 23 ; evidently the characteristic is 1.

In the narrow vertical column on the extreme left of the table, we see integers starting from 23, if we move

across horizontally, the figure just below 0 of the central column is 3617.