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  • Logarithms of Numbers Less Than One – (Introduction | Revision)

The logarithm of numbers less than one can be found using the negative power of 10 as the component characteristic.

For example, 0.001634

In standard form = 1.634 × 10-3

logarithims of numbers less than one e1668002514624

from tables = 100.2133 × 10-3

Using laws of indices = 100.2133 + (-3)

= 10-3+0.2133

Therefore we can say log100.001634 = -3 + 0.2133

The standard way of writing this is:

log100.001634 = \( \scriptsize \bar{3}.2133 \)

This reads ‘bar 3 point 2133’

Note that \( \scriptsize \bar{3}.2133 \) has a negative characteristic (-3) and positive mantissa (+ 0.2133)

Thus;

log100.1634 = \( \scriptsize \bar{1}.2133 \)
log100.01634 = \( \scriptsize \bar{2}.2133 \)
log100.001634 = \( \scriptsize \bar{3}.2133 \)
log100.0001634 = \( \scriptsize \bar{4}.2133 \)

Example 4.1.1:

Determine the characteristics of the logarithm of the numbers below:
(a) 0.002
(b) 0.1934
(c) 0.0000456
(d) 0.000789

Solution

(a) 0.002
Standard form: \( \scriptsize 2 \: \times \: 10^{-3} \)
Characteristic: \( \scriptsize \bar{3} \)

(b) 0.1934
Standard form: \( \scriptsize 1.934 \: \times \: 10^{-1} \)
Characteristic: \( \scriptsize \bar{1} \)

(c) 0.0000456
Standard form: \( \scriptsize 4.56 \: \times \: 10^{-5} \)
Characteristic: \( \scriptsize \bar{5} \)

(d) 0.000789
Standard form: \( \scriptsize 7.89 \: \times \: 10^{-4} \)
Characteristic: \( \scriptsize \bar{4} \)

Example 4.1.2:

Use the tables to find the logarithm of the following numbers:
(a) 0.006789
(b) 0.03498
(c) 0.007741
(d) 0.1265

Solution

 

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