Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: First Term
Week: 4
Age: 15 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Further Mathematics
Previous Lesson: Indices.
Topic: LOGARITHMS
Subject Matter: laws of logarithms, change of base of logarithms, solving logarithmic problems

Specific Objectives

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

Cognitive Domain:

Define a logarithm.
State the laws of logarithms.
Explain the concept of change of base of logarithms.
Solve problems involving laws of logarithms and change of base.

Affective Domain:

Appreciate the application of logarithms in solving mathematical problems.
Show interest in solving complex logarithmic equations.

Psychomotor Domain:

Accurately apply the laws of logarithms to simplify expressions.
Correctly use the change of base formula to evaluate logarithms.

Collaborate with peers to solve logarithmic problems.
Communicate mathematical ideas clearly when explaining solutions.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum for Senior Secondary Schools.
State Unified Scheme of Work for Further Mathematics SSS 1.
New Further Mathematics Project 1 for Senior Secondary Schools.

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts illustrating logarithm laws.
Illustrations demonstrating change of base.
Whiteboard and markers.

Rationale for the Lesson

This lesson helps pupils to understand logarithms as an important tool for solving complex mathematical problems, especially in areas like science and engineering. Understanding logarithms enables pupils to simplify calculations and work with very large or very small numbers effectively.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of indices, basic algebraic operations, and solving simple equations.

Lesson Content/Board Summary

LOGARITHMS

Definition of Logarithm

A logarithm is the power to which a base must be raised to produce a given number. If b^y = x, then y is the logarithm of x to the base b, written as log_b x = y.

Laws of Logarithms

The following are the fundamental laws of logarithms:

Product Law: log_b (MN) = log_b M + log_b N
Quotient Law: log_b (M/N) = log_b M – log_b N
Power Law: log_b (M^p) = p log_b M
Identity Law: log_b b = 1
Zero Law: log_b 1 = 0

Change of Base of Logarithms

Sometimes, it is necessary to convert a logarithm from one base to another. This is particularly useful when working with calculators that only have log (base 10) and ln (base e) functions.

The formula for changing the base of a logarithm from base ‘a’ to base ‘b’ is:

log_a x = (log_b x) / (log_b a)

A common application is changing to base 10 or base e:

log_a x = (log x) / (log a) (where ‘log’ implies base 10)
log_a x = (ln x) / (ln a) (where ‘ln’ implies base e)

Solving Logarithmic Problems

Solving logarithmic problems often involves applying the laws of logarithms to simplify expressions or equations, and sometimes using the change of base formula. The goal is usually to isolate the variable or simplify the expression to a numerical value.

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, reviews the previous lesson on indices, and asks questions to link indices to logarithms, such as “What power must 2 be raised to get 8?”. The teacher then introduces logarithms as the inverse operation of exponentiation.
Pupils’ Activity: Pupils respond to questions and listen attentively.
Learning Point: Pupils recall prior knowledge of indices and are introduced to the concept of logarithms.

Step 2: Presentation of Laws of Logarithms

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher writes the definition of logarithm on the board and explains each of the laws of logarithms using the charts. The teacher provides simple examples for each law.
Pupils’ Activity: Pupils listen, observe the charts, take notes, and ask questions for clarification.
Learning Point: Pupils understand the definition of logarithm and learn the various laws of logarithms.

Step 3: Explanation of Change of Base

Time: 7 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher explains the concept of changing the base of a logarithm, highlighting its importance for calculations using standard calculators. The teacher writes the change of base formula on the board and demonstrates its application with an example like calculating log₂ 5 using base 10.
Pupils’ Activity: Pupils listen, observe the illustrations, copy the formula, and follow the example.
Learning Point: Pupils understand the formula and application of changing the base of a logarithm.

Step 4: Worked Examples

Time: 8 minutes
Teaching Skill: Demonstration/Problem Solving
Teacher’s Activity: The teacher solves several problems on the board that involve applying a combination of the laws of logarithms and the change of base formula. For example: “Simplify log₃ 27 + log₃ 9” or “Evaluate log₄ 64”.
Pupils’ Activity: Pupils observe the steps, ask questions, and copy the worked examples into their notebooks.
Learning Point: Pupils learn how to apply the laws and change of base to solve problems.

Step 5: Guided Practice

Time: 6 minutes
Teaching Skill: Coaching/Facilitation
Teacher’s Activity: The teacher gives a few practice problems to the pupils and guides them through the solutions, providing hints and correcting mistakes as needed. For example: “Simplify log₂ (16/4)”.
Pupils’ Activity: Pupils attempt to solve the problems with guidance from the teacher and their peers.
Learning Point: Pupils practice applying the learned concepts with teacher support.

Step 6: Independent Practice

Time: 7 minutes
Teaching Skill: Independent Practice/Monitoring
Teacher’s Activity: The teacher assigns a few problems for pupils to solve individually or in small groups. The teacher circulates around the classroom to monitor progress and provide individual assistance.
Pupils’ Activity: Pupils work independently or collaboratively to solve the assigned problems.
Learning Point: Pupils consolidate their understanding and build confidence in solving logarithmic problems.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

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

Define a logarithm.
State any three laws of logarithms.
Explain the formula for changing the base of a logarithm.
Simplify log₂ 16 + log₂ 8.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization/Consolidation
Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the definition of logarithms, the laws, and the change of base formula. The teacher gives homework related to solving more logarithmic problems.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: Pupils consolidate their learning and receive further practice opportunities.

Lesson Keywords

Logarithm – The power to which a base must be raised to produce a given number.
Base – The number that is raised to a power in a logarithm.
Laws of Logarithms – Rules that govern how logarithmic expressions can be manipulated.
Change of Base – A formula used to convert a logarithm from one base to another.

Differentiation

For pupils who grasp the concepts quickly, the teacher will provide more challenging problems involving logarithmic equations. For pupils who need more support, the teacher will offer simplified examples and provide one-on-one guidance during practice sessions.

Note for teachers using this lesson plan

Ensure that pupils have a strong foundation in indices before introducing logarithms. Emphasize the practical applications of logarithms to keep pupils engaged. Encourage pupils to memorize the laws of logarithms as they are fundamental to solving problems efficiently.