LOGARITHMS

Contents:

1. Logarithm of numbers less than one, involving: Multiplication, Division, Powers and roots.
2. Solution of simple logarithmic equations.

Logarithm of Numbers Less than One

Simple Logarithm Operations

To find the logarithms of numbers less than 1, (i.e. numbers between 0 and 1), we use negative powers of 10.

For example, \(0.08356 = 8.356 × 10^{-2}\) (standard form)

\(0.08356 = 10^{0.9220} × 10^{-2}\) (from log tables)

\(= 10^{-2 + 0.9220} \)

So \(Log0.08356 = -2 + 0.9220\)

Characteristics (i.e. power) of \(10 = -2 \)

Mantissa \(= 0.9220\)

∴ \(-2 + 0.9220 = \bar{2}.9220\)

Note: −2 is called bar 2  i.e. \(\bar{2}\)

Example 1: Work out the following giving the answers in bar notation

(a) \(\bar{4}.3 × 5\)

(b) \(\bar{1}.6043 × 4\)

SOLUTION

(a) \(\bar{4}.3 × 5 = (\bar{4} + 0.3)5 \\ = \bar{20} + 1.5 \\ = \bar{19}.5 \)

(b) \(\bar{1}.6043 × 4 = \begin{array}{@{}rrr} & \bar{1} + 0.6043 \\ × & 4 \\ \hline & \bar{4} + 2.4172 \\ \hline \end{array} \\ ⇒ \bar{2}.4172\)

Example 2: Work out the following giving the answers in bar notation

\(\bar{5}.806 ÷ 4 \)

SOLUTION

\(\bar{5}.806 ÷ 4 = \frac{\bar{5} + 0.806}{4} \\ = \frac{\bar{8} + 3.806}{4} \\ = \bar{2} + 0.9515 \\ = \bar{2}.9515\)

CLASS ACTIVITY

Work out the following in bar notation form

(a)

(i) \(\begin{array}{@{}rrr} & \bar{3}.7 \\ + & \bar{5}.8 \\ \hline  \\ \hline \end{array}\)

(ii)  \(\begin{array}{@{}rrr} & \bar{2}.9 \\ + & 5.6 \\ \hline  \\ \hline \end{array}\)

(iii)  \(\begin{array}{@{}rrr} & \bar{5}.7 \\ − & \bar{2}.3 \\ \hline  \\ \hline \end{array}\)

(iv) \(\begin{array}{@{}rrr} & \bar{2}.8 \\ − & \bar{6}.1 \\ \hline  \\ \hline \end{array}\)

(v)  \(\begin{array}{@{}rrr} & \bar{5}.3 \\ − & \bar{2}.7 \\ \hline  \\ \hline \end{array}\)

(b) (i) \(\bar{3}.4 × 5\)

(ii) \(\bar{2}.823 × 4\)

(iii) \(\bar{3}.7538 ÷ 5\)

(iv) \(\bar{6}.509 ÷ 5\)

Logarithm of Numbers Less than One, Involving: Multiplication, Division, Powers and Roots

Example 1: (a) Evaluate, using logarithm tables \(0.9807 × 0.007692 \)

Solution:

\(0.9807 × 0.007692 \)

[missing table id=”13721″]

∴ Antilog of \(\bar{3}.8775 = 0.00754 \) (to 3 s.f.)

(b) Evaluate the following using logarithm tables \(0.00889 ÷ 204.6\)

[missing table id=”13722″]

∴ Antilog of \(\bar{5}.6380 = 0.0000435 \) (to 3 s.f.)

Note: In Logarithm, powers take multiplication while roots take division.