Lesson Note on Logarithms: Tables, Characteristics and Simple Equations for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 1st Term
Week: 1 – 2
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Number and Numeration
Previous Lesson: .
Topic: LOGARITHMS
Subject Matter: Revision of logarithm of numbers greater than one, characteristics of logarithm of numbers greater than one and less than one and standard form of numbers, logarithm of numbers less than one, multiplication, division, power and roots, solution of simple logarithmic equations, accuracy of results of logarithm table and calculator.

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• Recall the fundamental laws of logarithms (product, quotient, power, and change of base rules).
• Determine the characteristics of logarithms of numbers greater than one and less than one.
• Convert numbers to standard form and relate it to the characteristic of their logarithms.
• Perform calculations involving multiplication, division, powers, and roots using logarithms.
• Solve simple equations involving logarithms.

Affective Domain:

• Appreciate the importance of logarithms in simplifying complex calculations.
• Develop an interest in solving mathematical problems using logarithmic principles.
• Show diligence in checking the accuracy of results obtained from logarithm tables and calculators.

Psychomotor Domain:

• Accurately read and extract values from logarithm and antilogarithm tables.
• Correctly apply the laws of logarithms to solve numerical problems.
• Use a calculator efficiently for logarithmic computations.

Social Domain:

• Collaborate with peers to discuss and solve logarithmic problems.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• General Mathematics for Senior Secondary Schools by A.O. Salami et al.
• New General Mathematics for Senior Secondary Schools by J.B. Channon et al.

Instructional Materials

The teacher will teach this lesson with the aid of:

• Logarithm table
• Booklet on logarithm rules
• Solution chart of logarithm
• Flex banner showing logarithms and antilogarithm of numbers
• Calculator

Rationale for the Lesson

Understanding logarithms helps pupils simplify complex arithmetic calculations, especially those involving large numbers or numbers with many decimal places. It enables pupils to solve various mathematical problems and applies to fields like science, engineering, and finance.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of basic arithmetic operations, indices, and writing numbers in standard form.

Lesson Content/Board Summary

Logarithms

1. Laws of Logarithms

Logarithms are used to simplify calculations. The following are the fundamental laws:

• Product Law: log (xy) = log x + log y
• Quotient Law: log (x/y) = log x – log y
• Power Law: log (xⁿ) = n log x
• Change of Base Law: log_b a = log_c a / log_c b
• log_a a = 1
• log_a 1 = 0

2. Standard Form and Characteristics of Logarithms

A number is in standard form when written as A x 10ⁿ, where 1 ≤ A < 10 and n is an integer.

The characteristic of a logarithm is the integral part of the logarithm, while the mantissa is the positive decimal part.

• For numbers greater than one (N > 1): The characteristic is positive and is equal to (number of digits before the decimal point – 1).
  
  Example: For log 345.6, the characteristic is 3 – 1 = 2.
• For numbers less than one (N < 1): The characteristic is negative and is equal to -(number of leading zeros after the decimal point + 1). It is written with a bar over the number.
  
  Example: For log 0.0078, the characteristic is -(2 + 1) = -3, written as 3.

3. Calculations Involving Logarithms

Logarithms can be used to perform multiplication, division, powers, and roots.

Example 1: Multiplication and Division

Evaluate 25.34 x 0.0456 / 8.765

Solution:

Let P = 25.34 x 0.0456 / 8.765

Step 1: Take log of both sides: log P = log (25.34 x 0.0456 / 8.765)

Step 2: Apply logarithm laws:

log P = log 25.34 + log 0.0456 – log 8.765

Step 3: Find the logarithms from the table:

• log 25.34 = 1.4038
• log 0.0456 = 2.6590
• log 8.765 = 0.9428

Step 4: Perform addition and subtraction:

log P = (1.4038 + 2.6590) – 0.9428

log P = (1 + (-2) + 0.4038 + 0.6590) – 0.9428

log P = (-1 + 1.0628) – 0.9428

log P = 0.0628 – 0.9428

log P = 1.1200

Step 5: Find the antilogarithm:

P = antilog (1.1200) = 0.1318

Example 2: Powers and Roots

Evaluate (3.45)³ x ³√0.00789

Solution:

Let Q = (3.45)³ x ³√0.00789

Step 1: Take log of both sides: log Q = log [(3.45)³ x (0.00789)^1/3]

Step 2: Apply logarithm laws:

log Q = 3 log 3.45 + (1/3) log 0.00789

Step 3: Find the logarithms from the table:

• log 3.45 = 0.5378
• log 0.00789 = 3.8971

Step 4: Perform multiplication:

• 3 log 3.45 = 3 x 0.5378 = 1.6134
• (1/3) log 0.00789 = (1/3) x 3.8971

Note: For negative characteristics, make the characteristic divisible by the denominator of the fraction. Add and subtract a number to achieve this.

(1/3) x (3.8971) = (1/3) x (3 + 0.8971) = 1 + 0.2990 = 1.2990

Step 5: Add the results:

log Q = 1.6134 + 1.2990

log Q = (1 + (-1)) + (0.6134 + 0.2990)

log Q = 0 + 0.9124 = 0.9124

Step 6: Find the antilogarithm:

Q = antilog (0.9124) = 8.174

4. Solving Simple Logarithmic Equations

Logarithmic equations can be solved by applying the laws of logarithms and converting to index form where necessary.

Example 3:

Solve for x: log x + log 4 = log 12

Solution:

Step 1: Apply product law: log (x * 4) = log 12

Step 2: Simplify: log (4x) = log 12

Step 3: Equate the numbers (since logs are equal): 4x = 12

Step 4: Solve for x: x = 12 / 4 = 3

Example 4:

Solve for x: 2 log x = log 25

Solution:

Step 1: Apply power law: log x² = log 25

Step 2: Equate the numbers: x² = 25

Step 3: Solve for x: x = √25 = 5 (since logarithm is defined for positive numbers, we take the positive root).

5. Accuracy of Results

Results obtained using logarithm tables are approximations due to rounding. Calculators generally provide more accurate results as they perform calculations with higher precision. It is important to understand the limitations of each method.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher greets the pupils and reminds them about their previous knowledge of indices, explaining that logarithms are closely related to indices. The teacher asks pupils to recall the meaning of indices like 2³.
Pupils’ Activity: Pupils respond to greetings and recall the meaning of indices such as 2³ = 8.
Learning Point: Pupils establish a connection between indices and logarithms.

Step 2: Revision of Laws of Logarithms

Time: 10 minutes
Teaching Skill: Explaining/Recalling
Teacher’s Activity: The teacher revises the product, quotient, power, and change of base laws of logarithms, providing simple examples for each. The teacher also explains log_a a = 1 and log_a 1 = 0.
Pupils’ Activity: Pupils state the laws of logarithms and participate in solving simple examples.
Learning Point: Pupils recall and understand the fundamental laws of logarithms.

Step 3: Standard Form and Characteristics of Logarithms

Time: 7 minutes
Teaching Skill: Explaining/Illustrating
Teacher’s Activity: The teacher explains how to write numbers in standard form and demonstrates how to determine the characteristic of logarithms for numbers greater than one and less than one, showing their relationship to standard form.
Pupils’ Activity: Pupils write given numbers in standard form and determine their characteristics.
Learning Point: Pupils understand the relationship between standard form and the characteristic of a logarithm.

Step 4: Logarithms of Numbers Greater Than One (Calculations)

Time: 7 minutes
Teaching Skill: Demonstrating/Calculating
Teacher’s Activity: The teacher demonstrates how to use logarithm tables and calculators to perform calculations involving multiplication, division, powers, and roots of numbers greater than one, using worked examples.
Pupils’ Activity: Pupils follow the teacher’s examples, read logarithm tables, and perform calculations.
Learning Point: Pupils learn to apply logarithm laws to calculate numbers greater than one.

Step 5: Logarithms of Numbers Less Than One (Calculations)

Time: 7 minutes
Teaching Skill: Demonstrating/Problem-solving
Teacher’s Activity: The teacher demonstrates how to handle logarithms of numbers less than one, especially in calculations involving multiplication, division, powers, and roots, providing clear steps for managing negative characteristics.
Pupils’ Activity: Pupils practice calculations involving numbers less than one, paying attention to the handling of negative characteristics.
Learning Point: Pupils develop skills in calculating with logarithms of numbers less than one.

Step 6: Solution of Simple Logarithmic Equations

Time: 5 minutes
Teaching Skill: Problem-solving/Guiding
Teacher’s Activity: The teacher guides pupils through solving simple logarithmic equations by applying the laws of logarithms and converting to index form where appropriate.
Pupils’ Activity: Pupils attempt to solve simple logarithmic equations under the teacher’s guidance.
Learning Point: Pupils learn to solve basic logarithmic equations.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. State any three laws of logarithms.
2. What is the characteristic of log 678.9?
3. What is the characteristic of log 0.0054?
4. Simplify log 6 + log 5 – log 3.
5. Solve for x: log x = 3.2456.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 4 minutes
Teaching Skill: Summarizing/Assigning
Teacher’s Activity: The teacher summarizes the key learning points on logarithms, emphasizing the importance of understanding the laws and the characteristic. The teacher assigns homework from the textbook.
Pupils’ Activity: Pupils pay attention to the summary and copy down the assigned homework.
Learning Point: Pupils reinforce their understanding and prepare for independent practice.

Lesson Keywords

• Logarithm – The power to which a base number must be raised to yield a given number.
• Characteristic – The integral part of a logarithm.
• Mantissa – The positive decimal part of a logarithm, obtained from logarithm tables.
• Standard Form – Writing a number as A x 10ⁿ, where 1 ≤ A < 10.
• Antilogarithm – The number whose logarithm is a given value.

Differentiation

For struggling learners, the teacher will provide simplified examples and extra practice using the logarithm table, focusing on one law at a time. Advanced learners will be given more complex problems involving multiple laws and real-world applications of logarithms, encouraging them to explore the use of scientific calculators for higher accuracy.

Note for teachers using this lesson plan

Ensure pupils have access to logarithm tables or scientific calculators. Emphasize the systematic approach to solving problems, especially when dealing with negative characteristics. Encourage pupils to check their answers using both methods (table and calculator) to understand accuracy differences.