Logarithms, Deducing from Indices and Standard Form for SS 1

Class: SS 1
Term: First Term
Week: 7
Age: 15 years
Duration: 60 minutes
Subject: General Mathematics
Curriculum Theme: Number and Numeration
Focal competence: Changing Indices notations and potting graph of y=10
Key competencies/values: ICT and Digital Competencies; Digital Competencies

Skills:

• Plotting Graph of y=10

Previous Lesson: Application of trigonometric ratios of special angles to simple problems
Topic: Logarithms
Subject Matter: Deducing logarithm from indices and standard form, Definition of logarithm

Specific Objectives

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

Cognitive Domain

• Define logarithms and antilogarithms.
• Change indices notations to logarithm (log) notations.

Psychomotor Domain

• Plot the graph of (y = 10^x).

Reference Materials

The following resources were used in planning this lesson:

• 2025 New Revised Senior Secondary Education Curriculum (SSEC)
• Relevant State Unified Scheme of Work
• A suitable Mathematics textbook for SS 1
• The HeadTeacher Scheme of work For The New Revised Senior Secondary Education Curriculum (SSEC)

Instructional Materials

The teacher will teach this lesson with the aid of:

• Indices chart
• Logarithms chart
• Graph board with graph of (y = 10^x)
• Graph book
• Video clip to find the Log and antilog of a given number
• Mathematical tables (Logarithm tables)

Rationale for the Lesson

This lesson is important because logarithms provide a powerful tool for simplifying complex calculations and solving exponential equations, which are common in science, engineering, and finance. Understanding logarithms also deepens students’ comprehension of number relationships and prepares them for advanced mathematical concepts.

Prerequisite/Previous Knowledge

Students should have a good understanding of indices (exponents) and standard form of numbers.

Lesson Content/Board Summary

Logarithms

Definition of Logarithm

A logarithm is the power or exponent to which a fixed number, called the base, must be raised to produce a given number. In simpler terms, it answers the question: “How many of one number do we multiply to get another number?”

For example, since (10^2 = 100), the logarithm of 100 to base 10 is 2. This is written as (log_{10} 100 = 2).

Generally, if (b^x = N), then (x = log_b N), where:

1. (b) is the base (must be a positive number not equal to 1).
2. (x) is the logarithm (the exponent).
3. (N) is the number (must be a positive number).

Relationship between Indices and Logarithms

Logarithms are essentially the inverse operation of exponentiation. Every exponential statement can be rewritten in logarithmic form, and vice versa.

Converting from Index Form to Logarithmic Form

To convert an expression from index form (b^x = N) to logarithmic form (x = log_b N), follow these steps:

1. Identify the base ((b)).
2. Identify the exponent ((x)).
3. Identify the result ((N)).
4. Write it as “(log_{text{base}} text{result} = text{exponent})”.

Examples:

1. If (2^3 = 8), then (log_2 8 = 3).
2. If (10^4 = 10000), then (log_{10} 10000 = 4).
3. If (5^2 = 25), then (log_5 25 = 2).
4. If (3^{-2} = frac{1}{9}), then (log_3 frac{1}{9} = -2).

Converting from Logarithmic Form to Index Form

To convert an expression from logarithmic form (x = log_b N) to index form (b^x = N), follow these steps:

1. Identify the base ((b)).
2. Identify the logarithm ((x), which is the exponent).
3. Identify the number ((N)).
4. Write it as “(text{base}^{text{exponent}} = text{number})”.

Examples:

1. If (log_2 16 = 4), then (2^4 = 16).
2. If (log_{10} 1000 = 3), then (10^3 = 1000).
3. If (log_7 49 = 2), then (7^2 = 49).
4. If (log_4 frac{1}{4} = -1), then (4^{-1} = frac{1}{4}).

Logarithms and Standard Form

Standard form (or scientific notation) expresses a number as (A times 10^n), where (1 le A < 10) and (n) is an integer. Logarithms to base 10 are closely related to standard form.

When we find the logarithm of a number to base 10, the result consists of two parts:

1. Characteristic: The integer part of the logarithm. This is determined by the power of 10 in the standard form of the number.
2. Mantissa: The decimal part of the logarithm. This is always positive and is found using logarithm tables.

If a number (N) is written in standard form as (N = A times 10^n), then (log_{10} N = log_{10} (A times 10^n) = log_{10} A + log_{10} 10^n = log_{10} A + n).

Here, (n) is the characteristic and (log_{10} A) is the mantissa (since (1 le A < 10), (0 le log_{10} A < 1)).

Example:

Consider the number 3450.

1. Write 3450 in standard form: (3.45 times 10^3). Here, (n=3).
2. The characteristic of (log_{10} 3450) is 3.
3. Using a logarithm table, (log_{10} 3.45 approx 0.5378).
4. So, (log_{10} 3450 approx 3.5378).

Antilogarithms

The antilogarithm (antilog) is the inverse operation of the logarithm. If (x = log_b N), then (N) is the antilogarithm of (x) to the base (b).

In other words, if you are given the logarithm of a number and you want to find the original number, you find its antilogarithm. This is equivalent to raising the base to the power of the given logarithm.

If (log_{10} N = x), then (N = 10^x).

Example:

1. If (log_{10} N = 2), then (N = text{antilog } 2 = 10^2 = 100).
2. If (log_{10} N = 3.5378), then (N = text{antilog } 3.5378 = 10^{3.5378} approx 3450).

Graph of (y = 10^x)

The graph of (y = 10^x) is an exponential curve. It shows the relationship between a number (x) (the exponent) and its corresponding value (y) (the result when 10 is raised to that power).

Key features of the graph (y = 10^x):

1. It always passes through the point ((0, 1)) because (10^0 = 1).
2. As (x) increases, (y) increases rapidly (exponential growth).
3. As (x) decreases, (y) approaches the x-axis but never touches it (the x-axis is an asymptote).
4. All values of (y) are positive.

This graph is crucial because its inverse, (x = log_{10} y), represents the logarithm function. The graph visually demonstrates how a small change in (x) can lead to a large change in (y), and vice versa for the inverse logarithm function.

Teaching Methods/Instructional Techniques

Explanation, Discussion, Demonstration, Guided Practice, Question and Answer, Problem Solving

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Review/Connection

Teacher’s Activity: The teacher revises previous knowledge on indices and standard form by asking questions like, “What is (2^3)?”, “How do you write 5000 in standard form?”. The teacher then introduces the topic of logarithms as an inverse way of expressing indices.

Pupils’ Activity: Pupils answer questions on indices and standard form, recalling prior knowledge.

Learning Point: Recalling indices and standard form

Step 2: Defining Logarithm

Time: 10 minutes

Teaching Skill: Explanation/Definition

Teacher’s Activity: The teacher defines logarithm, explaining it as the power to which a base must be raised to get a number. The teacher provides examples like (10^2 = 100) means (log_{10} 100 = 2).

Pupils’ Activity: Pupils listen attentively, ask questions for clarification, and write down the definition.

Learning Point: Meaning of logarithm

Step 3: Relationship between Indices and Logarithms

Time: 10 minutes

Teaching Skill: Demonstration/Explanation

Teacher’s Activity: The teacher uses the indices chart to demonstrate how index notations can be changed to logarithm notations and vice versa. The teacher writes several examples on the board and explains the process step-by-step.

Pupils’ Activity: Pupils observe the chart, follow the teacher’s explanation, and participate in converting simple examples.

Learning Point: Index-logarithm conversion

Step 4: Practice Converting Notations

Time: 10 minutes

Teaching Skill: Guided Practice

Teacher’s Activity: The teacher provides more examples for students to practice changing from index form to logarithm form and from logarithm form to index form. The teacher moves around to assist students.

Pupils’ Activity: Pupils work on the given examples in their notebooks, converting between the two forms.

Learning Point: Converting index and log forms

Step 5: Logarithms and Standard Form

Time: 5 minutes

Teaching Skill: Explanation/Connection

Teacher’s Activity: The teacher explains how logarithms to base 10 relate to numbers in standard form, introducing the terms ‘characteristic’ and ‘mantissa’. The teacher gives a simple example to illustrate the characteristic.

Pupils’ Activity: Pupils listen and relate the concept to their knowledge of standard form.

Learning Point: Logarithms and standard form

Step 6: Antilogarithms and Graph of (y = 10^x)

Time: 5 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher defines antilogarithm as the inverse of logarithm. The teacher then uses the graph board or a projected image to show the graph of (y = 10^x), explaining its features and its connection to logarithms.

Pupils’ Activity: Pupils listen to the definition of antilogarithm and observe the graph, understanding its significance.

Learning Point: Antilogarithm and exponential graph

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. Define logarithm.
2. Change (3^4 = 81) to logarithm notation.
3. Change (log_5 125 = 3) to index notation.
4. What is an antilogarithm?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Understanding logarithm concepts

Step 8: Note-Taking

Time: 10 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes on logarithms, their definition, and conversion between forms into their notebooks.

Pupils’ Activity: Pupils/students copy the notes carefully into their notebooks.

Learning Point: Recording lesson notes

Step 9: Conclusion

Time: 0 minutes

Teaching Skill: Consolidation

Teacher’s Activity: The teacher briefly reminds students that logarithms are simply another way of expressing indices and are very useful in simplifying large numbers. The teacher encourages them to practice more examples.

Pupils’ Activity: Pupils listen and prepare for the next lesson.

Learning Point: Logarithm concept reinforced

Continuous Assessment/Further Study

Type: Homework

Instruction: Answer the following questions in your notebook.

1. Define logarithm and antilogarithm in your own words.
2. Convert the following index notations to logarithm notations:
   1. (4^3 = 64)
   2. (7^2 = 49)
   3. (10^{-1} = 0.1)
3. Convert the following logarithm notations to index notations:
   1. (log_6 36 = 2)
   2. (log_2 32 = 5)
   3. (log_{10} 100000 = 5)
4. Explain how the characteristic of a base 10 logarithm relates to the standard form of a number.

Lesson Keywords

• Logarithm – The power to which a base must be raised to produce a given number.
• Base – The fixed number that is raised to an exponent in an exponential expression or logarithm.
• Exponent/Index – The power to which a number is raised.
• Antilogarithm – The number corresponding to a given logarithm.
• Characteristic – The integer part of a logarithm.
• Mantissa – The positive decimal part of a logarithm.
• Standard Form – A way of writing numbers using powers of ten.

Differentiation

Support: Provide simpler examples and more direct guidance for students struggling with the conversion between index and logarithm forms. Use visual aids like number lines or charts to illustrate the concept of powers.

Extension: Challenge advanced students to explore logarithms with different bases (e.g., base 2 or base 5) or to investigate the basic laws of logarithms (e.g., product rule, quotient rule) as an introduction to future topics.

Suggested Lesson Videos

Search YouTube for: “Introduction to Logarithms SS1 Mathematics” or “Logarithms from Indices and Standard Form”