LOGARITHMS
Contents:
- Logarithm of numbers less than one, involving: Multiplication, Division, Powers and roots.
- 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.
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