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

from tables = 10^0.2133 × 10^-3

Using laws of indices = 10^{0.2133 + (-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 1.1.1:

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 1.1.2:

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

Solution

(a) 0.006789
= \( \scriptsize \log 6.789 \: \times \: 10^{-3} \)
= \( \scriptsize 10^{0.8318} \: \times \: 10^{-3} \)
= \( \scriptsize 10^{-3 \: + \: 0.8318} \)
= \( \scriptsize \bar{3}.8318 \)

(b) 0.03498
= \( \scriptsize \log 3.498 \: \times \: 10^{-2} \)
= \( \scriptsize 10^{0.5438} \: \times \: 10^{-2} \)
= \( \scriptsize 10^{-2 \: + \: 0.5438} \)
= \( \scriptsize \bar{2}.5438 \)

(c) 0.007741
= \( \scriptsize \log 7.741 \: \times \: 10^{-3} \)
= \( \scriptsize 10^{0.8888} \: \times \: 10^{-3} \)
= \( \scriptsize 10^{-3 \: + \: 0.8888} \)
= \( \scriptsize \bar{3}.8888 \)

(d) 0.1265
= \( \scriptsize \log 1.265 \: \times \: 10^{-1} \)
= \( \scriptsize 10^{0.1021} \: \times \: 10^{-1} \)
= \( \scriptsize 10^{-1 \: + \: 0.1021} \)
= \( \scriptsize \bar{1}.1021 \)

Example 1.1.3:

Use the anti-log tables to find the number whose logarithm (base 10) is
(a) 0.3467
(b) 1.3467
(c) \( \scriptsize \bar{2}.1021 \)
(d) \( \scriptsize \bar{4}.1021 \)

Solution

(a) 0.3467
let the number be x
log_x = 0.3467
x = antilog of 0.3467
x = 10^0.3467
x = 10^{0 + 0.3467}
x = 10⁰ × 10^0.3467
x = 1 × 2.222
x = 2.222

(b) 1.3467
let the number be x
log_x = 1.3467
x = antilog of 1.3467
x = 10^1.3467
x = 10^{1 + 0.3467}
x = 10¹ × 10^0.3467
x = 10 × 2.222
x = 22.22

(c) \( \scriptsize \bar{2}.1021 \)
let the number be x
log_x = \( \scriptsize \bar{2}.1021 \)
x = antilog of \( \scriptsize \bar{2}.1021 \)
x = \( \scriptsize 10^{\bar{2}.1021} \)
x = \( \scriptsize 10^{\bar{2} \: + \: 0.1021} \)
x = \( \scriptsize 10^{\bar{2}} \: \times \: 10^{0.1021} \)
x = \( \scriptsize 10^{-2} \: \times \: 10^{0.1021} \)
x = \( \frac{1}{10^2} \scriptsize \: \times \: 10^{0.1021} \)

x = \( \frac{1}{100} \scriptsize \: \times \: 1.256 \)

x = \( \scriptsize 0.01 \: \times \: 1.265 \)

x = \( \scriptsize 0.01265 \)

(d) \( \scriptsize \bar{4}.1021 \)
let the number be x
log_x = \( \scriptsize \bar{4}.1021 \)
x = antilog of \( \scriptsize \bar{4}.1021 \)
x = \( \scriptsize 10^{\bar{4}.1021} \)
x = \( \scriptsize 10^{\bar{4} \: + \: 0.1021} \)
x = \( \scriptsize 10^{\bar{4}} \: \times \: 10^{0.1021} \)
x = \( \scriptsize 10^{-4} \: \times \: 10^{0.1021} \)
x = \( \frac{1}{10^4} \scriptsize \: \times \: 10^{0.1021} \)

x = \( \frac{1}{10000} \scriptsize \: \times \: 1.256 \)

x = \( \scriptsize 0.0001 \: \times \: 1.265 \)

x = \( \scriptsize 0.0001265 \)