Logarithm

Mathematics SSS 2 First Term

Theme: Numbers and Numeration

WEEK 2

Logarithm

Performance Objectives

Students should be able to;

1. Calculate numbers less than 1 involving multiplication and division using log table
2. Solve simple logarithm equations

Revision of Logarithm Numbers Greater than 1

Content

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

Example 2: Evaluate using logarithm tablesAntilog of 4.0521=11300 to 3 s.f.

 

 

	Numbers	Log
	173.8	2.2400	2.2400
	(14.7)2	1.1673 × 2	+ 2.3346
		Numerator	4.5746		4.5746
	(2.61)3	0.4166 × 3	1.2498		- 1.2498
	173.8 x (14.7)2(2.61)3				3.3248
	173.8 x (14.7)2(2.61)3					3.3248 ÷ 2
	45.96					1.6624

∴ Antilog of 1.6624=45.96 (to 4 s.f.)

Logarithm of Numbers Less than One

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= 2.9220

Note: −2 is called bar 2  i.e. 2

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

(a) 4.3 × 5

(b) 1.6043 × 4

Solution

(a) 4.3 × 5 = (4 + 0.3)5 = 20 + 1.5 = 19.5

(b) 1.6043 × 4 = 1 + 0.6043x               44 + 2.4172   ⇒   2 . 4172

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

5.806÷4

Solution

5.806 ÷ 4= 5 + 0.8064 = 8 + 3.8064=2 + 0.9515=2.9515

Multiplication

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

Solution:

0.9807 × 0.007692

 

	Number	Log
	0.9807	1.9915
	0.007692	+ 3.8860
	0.007543	3.8775

∴ Antilog of 3.8775=0.00754 (to 3 s.f.)

Division

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

 

	Numbers	Log
	0.00889	3.9489
	204.6	-  2.3109
	0.00004345	5.6380

∴ Antilog of 5.6380=0.0000435 (to 3 s.f.)

Powers

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

Example 2: (a) Evaluate (0.05872)⁴

 

	Numbers	Log
	0.05872	2.7687
	0.058724	2.7687×4
	0.00001188	5.0748

∴ (0.05872)⁴ = 0.000012 (to 2 s.f.)

Roots

(b) 70.0004786

Solution:

 

	Numbers	Log
	0.0004786	4.6799
	70.0004786	4.6799 ÷ 7
		7+3.6799÷7
		1+0.5257
	0.3355	1.5257

∴ 70.0004786 = 0.3355

 

Solution of Simple Logarithmic Equations

In this lesson, the equations have to be solved first, then tables used to evaluate

If log_ay = x logay=x

Then y = a^x y=ax

i.e 100 = 10²  and log₁₀100 = 2

Then log to the base 10 of 100 is 2.  Logarithm can easily be in any base. For example since 64 = 4³, then log₄ 64 = 3 ( the log to the base 4 of 64 is 3).

Example

Find x if log_x 8 = 3

If log_x 8 = 3, then x³ = 8

Therefore x = 2