LOGARITHMS

CONTENT

1. Comparison of the Characteristics of Logarithms and Standard Form of Numbers
2. Revision of Logarithm Numbers Greater than 1

Comparison of Characteristics of Logarithms and Standard Form of Numbers

There is a relationship between the standard form and the logarithm of a number.

For instance, (a) \(189.7 = 1.897 × 10^2\)

This shows that the logarithm of a number is the power to which the base 10 is raised. Hence, Logarithm of \(189.7 = 2.2781,\) where \(189.7 = 10^{2.2781} \)

(b) \(850.9 = 8.509 × 10^2\) (standard form)

\(Log 850.9 = 2.9299\)

The integer (characteristics) is the same with the power 10

CLASS ACTIVITY

Show how the characteristics following Logarithms are related to standard form

1. 82000
2. 68.9
3. 6895
4. 605.8

Revision of Logarithm Numbers Greater than 1

Logarithm of numbers is the power to which 10 is raised to give that number. Logarithms used in calculations are normally expressed in base 10.

Rules for the Use of Logarithms

1. Multiplication: find the logarithms of the numbers and add them together
2. Division: find the logarithm of each number. Then subtract the logarithm of the denominator from that of the numerator
3. Powers: find the logarithm of the number and then multiply it by the power or the index
4. Roots: find the logarithm of the number and then divide it by the root

Example 1: Evaluate using logarithm tables

\(19.28 × 2.987 × 195.8\)

Antilog of \(4.0521 = 11300\) to 3 s.f.

Example 2: Evaluate using logarithm tables

\(\sqrt{\frac{173.8 × (14.7)^2 }{(2.61)^3}}\)

∴ Antilog of \(1.6624 = 45.96\) (to 4 s.f.)

CLASS ACTIVITY

Use logarithm tables to evaluate correct to 4 s.f.

1. \(786.1 × 89.5 × 63.7\)

2. \(\frac{107.8 × 38.97 }{81.65}\)

PRACTICE EXERCISES

Use log tables to find the value of the following

1. \(\frac{\sqrt{17.45} × (35.2)^2 }{(3.15)^4 × 8.15}\)

2. \(\frac{(27.1)^2 × 327}{\sqrt{27500000}}\)

3. \(\frac{95.3 × \sqrt[3]{18.4} }{(1.29)^5 × 2.03}\)

4. \(\frac{298.6 × 10.52 }{2.56 × 32.8}\)

5. \(\sqrt[3]{\frac{321000 × 40}{175 × 6000}}\)

ASSIGNMENT

Use log tables to find the value of the following

1.  \(\frac{6.705 × 3.68^3}{\sqrt[4]{35.81}}\)

2. \(\frac{23.67 × 73.59 }{(2.5134)^5 × 2.03}\)

3. \((987.3)^{\frac{1}{3}}\)

4. \(\sqrt[3]{\frac{875.4}{(4.234)^3}}\)

5. \(\sqrt{23.56 × 66.45}\)

KEYWORDS: base, logarithm, integer, antilogarithm, mantissa, characteristics etc.