Logarithms, Laws, Tables and Theory for SS 3

Class: Senior Secondary School 3 (SS3 / SSS3)
Term: 1st Term
Week: 1 – 2
Age: 17 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Number and Numeration
Previous Lesson:
Topic: LOGARITHMS
Subject Matter: Revision of law of indices, revision of the use of logarithm table to calculate logarithm of numbers, theory of logarithm, rules connecting logarithm Log(pq)=log p + log q, Log(p/q)=log p – log q, Log pⁿ = nlog p

Specific Objectives

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

Cognitive Domain:

• Recall the basic laws of indices.
• Define logarithm and state its relationship with indices.
• State the three basic laws of logarithm (product, quotient, and power laws).
• Explain how to use logarithm tables to perform calculations involving multiplication, division, powers, and roots.

Affective Domain:

• Appreciate the importance of logarithms in simplifying complex calculations.
• Develop an interest in applying logarithm laws to solve problems.

Psychomotor Domain:

• Apply the laws of logarithm to simplify expressions.
• Accurately use logarithm tables to find the logarithm and antilogarithm of numbers.
• Solve problems involving multiplication, division, powers, and roots using logarithm tables.

Social Domain:

• Participate actively in class discussions and problem-solving sessions.

Reference Materials

The following resources were used in planning this lesson:

• FCT-ERC SS 3 Scheme of Work, September 2014 curriculum.
• Relevant State Unified Scheme of Work.
• A suitable General Mathematics textbook for SS 3.

Instructional Materials

The teacher will teach this lesson with the aid of:

• Logarithm law chart
• Solution chart of logarithm
• Logarithm table
• Calculators (for verification)

Rationale for the Lesson

This lesson helps students understand how to simplify complex multiplication, division, power, and root problems using logarithms. Mastering logarithms enables students to solve advanced mathematical problems and is useful in various scientific and engineering fields.

Prerequisite/Previous Knowledge

Students are expected to have prior knowledge of basic arithmetic operations, laws of indices, and the concept of exponents from Junior Secondary School.

Lesson Content/Board Summary

LOGARITHMS

Revision of Laws of Indices

Indices (or exponents) represent the number of times a base number is multiplied by itself. For example, in aⁿ, ‘a’ is the base and ‘n’ is the index or exponent.

The following are fundamental laws of indices:

1. Product Law: a^m × aⁿ = a^m+n
2. Quotient Law: a^m ÷ aⁿ = a^m-n (where a ≠ 0)
3. Power Law: (a^m)ⁿ = a^mn
4. Zero Index Law: a⁰ = 1 (where a ≠ 0)
5. Negative Index Law: a^-n = 1/aⁿ (where a ≠ 0)
6. Fractional Index Law: a^1/n = ⁿ√a
7. Fractional Index Law: a^m/n = (ⁿ√a)^m

Example: Simplify 2³ × 2⁴.

Solution:

1. Step 1: Identify the law. This is the product law of indices (a^m × aⁿ = a^m+n).
2. Step 2: Apply the law. 2³ × 2⁴ = 2⁽³⁺⁴⁾
3. Step 3: Simplify. 2⁷ = 128.

Theory of Logarithm

A logarithm is the power to which a base must be raised to produce a given number. It is the inverse operation to exponentiation.

If a^x = N, then x = log_a N.

1. Here, ‘a’ is the base, ‘x’ is the logarithm, and ‘N’ is the number.
2. The expression ‘log_a N’ is read as “logarithm of N to base a”.
3. Common bases for logarithms are base 10 (common logarithm, often written as log N) and base e (natural logarithm, written as ln N).

Example: Convert 10² = 100 to logarithm form.

Solution:

1. Step 1: Identify base, power, and number. Base (a) = 10, Power (x) = 2, Number (N) = 100.
2. Step 2: Apply the logarithm definition. If a^x = N, then x = log_a N.
3. Step 3: Substitute values. 2 = log₁₀ 100.

Laws of Logarithm

Logarithm laws are derived from the laws of indices and are used to simplify logarithmic expressions and calculations.

The following are the main laws of logarithm:

1. Product Law: log_a (pq) = log_a p + log_a qThis law states that the logarithm of a product of two numbers is the sum of their individual logarithms to the same base.

   Example: Simplify log₁₀ (5 × 7).

   Solution:

   1. Step 1: Identify the law. This is the product law.
   2. Step 2: Apply the law. log₁₀ (5 × 7) = log₁₀ 5 + log₁₀ 7.
   3. Step 3: (If values are known) Calculate. (Values would be looked up in a log table if required for numerical answer).
2. Quotient Law: log_a (p/q) = log_a p – log_a qThis law states that the logarithm of a quotient of two numbers is the difference between the logarithm of the numerator and the logarithm of the denominator to the same base.

   Example: Simplify log₁₀ (80 / 10).

   Solution:

   1. Step 1: Identify the law. This is the quotient law.
   2. Step 2: Apply the law. log₁₀ (80 / 10) = log₁₀ 80 – log₁₀ 10.
   3. Step 3: Simplify. log₁₀ 80 – 1 (since log₁₀ 10 = 1).
3. Power Law: log_a pⁿ = n log_a pThis law states that the logarithm of a number raised to a power is the product of the power and the logarithm of the number to the same base.

   Example: Simplify log₁₀ 2³.

   Solution:

   1. Step 1: Identify the law. This is the power law.
   2. Step 2: Apply the law. log₁₀ 2³ = 3 log₁₀ 2.
   3. Step 3: (If values are known) Calculate. (Values would be looked up in a log table if required for numerical answer).

Other important laws:

1. log_a a = 1
2. log_a 1 = 0
3. Change of Base Law: log_a b = (log_c b) / (log_c a)

Using Logarithm Tables for Calculations

Logarithm tables are used to find the logarithm of numbers to base 10 (common logarithms) and to perform complex calculations.

Every logarithm (log₁₀ N) consists of two parts: the characteristic and the mantissa.

1. Characteristic: The integral part of the logarithm. It is determined by the position of the decimal point in the number.
   1. For numbers greater than or equal to 1, the characteristic is one less than the number of digits before the decimal point. (e.g., log 567.8 has characteristic 2).
   2. For numbers less than 1, the characteristic is negative and is one more than the number of zeros immediately after the decimal point. It is written with a bar over it. (e.g., log 0.0567 has characteristic ī.7536, where ī indicates -2).
2. Mantissa: The decimal part of the logarithm. It is always positive and is found from the logarithm table.

Steps for finding the logarithm of a number:

1. Determine the characteristic.
2. Find the mantissa from the logarithm table using the significant figures of the number.
3. Combine the characteristic and mantissa.

Steps for finding the antilogarithm of a number:

1. Identify the mantissa and characteristic from the given logarithm.
2. Find the number corresponding to the mantissa using the antilogarithm table.
3. Use the characteristic to place the decimal point in the number.

Calculations using Logarithm Tables:
1. Multiplication (using Product Law): To calculate A × B

Formula: log (A × B) = log A + log B

Steps:

1. Find log A.
2. Find log B.
3. Add log A and log B.
4. Find the antilogarithm of the sum.

Example: Calculate 25.4 × 3.12 using logarithm tables.

Solution:

1. Step 1: Find log 25.4. Characteristic = 1. Mantissa (from table for 254) = 0.4048. So, log 25.4 = 1.4048.
2. Step 2: Find log 3.12. Characteristic = 0. Mantissa (from table for 312) = 0.4942. So, log 3.12 = 0.4942.
3. Step 3: Add the logarithms.
   1.4048
   + 0.4942
   ———-
   1.8990
4. Step 4: Find the antilog of 1.8990. Antilog of 0.8990 (from antilog table) = 7.926.
   Characteristic = 1, so there are 1+1 = 2 digits before the decimal point.
   Therefore, Antilog (1.8990) = 79.26.

2. Division (using Quotient Law): To calculate A ÷ B

Formula: log (A ÷ B) = log A – log B

Steps:

1. Find log A.
2. Find log B.
3. Subtract log B from log A.
4. Find the antilogarithm of the difference.

Example: Calculate 78.5 ÷ 1.25 using logarithm tables.

Solution:

1. Step 1: Find log 78.5. Characteristic = 1. Mantissa (from table for 785) = 0.8949. So, log 78.5 = 1.8949.
2. Step 2: Find log 1.25. Characteristic = 0. Mantissa (from table for 125) = 0.0969. So, log 1.25 = 0.0969.
3. Step 3: Subtract the logarithms.
   1.8949
   – 0.0969
   ———-
   1.7980
4. Step 4: Find the antilog of 1.7980. Antilog of 0.7980 (from antilog table) = 6.281.
   Characteristic = 1, so there are 1+1 = 2 digits before the decimal point.
   Therefore, Antilog (1.7980) = 62.81.

3. Powers (using Power Law): To calculate Aⁿ

Formula: log (Aⁿ) = n log A

Steps:

1. Find log A.
2. Multiply log A by n.
3. Find the antilogarithm of the product.

Example: Calculate (3.5)² using logarithm tables.

Solution:

1. Step 1: Find log 3.5. Characteristic = 0. Mantissa (from table for 350) = 0.5441. So, log 3.5 = 0.5441.
2. Step 2: Multiply log 3.5 by 2.
   0.5441 × 2 = 1.0882
3. Step 3: Find the antilog of 1.0882. Antilog of 0.0882 (from antilog table) = 1.225.
   Characteristic = 1, so there are 1+1 = 2 digits before the decimal point.
   Therefore, Antilog (1.0882) = 12.25.

4. Roots: To calculate ⁿ√A

Formula: log (ⁿ√A) = log (A^1/n) = (1/n) log A

Steps:

1. Find log A.
2. Divide log A by n.
3. Find the antilogarithm of the quotient.

Example: Calculate √64 using logarithm tables.

Solution:

1. Step 1: Find log 64. Characteristic = 1. Mantissa (from table for 640) = 0.8062. So, log 64 = 1.8062.
2. Step 2: Divide log 64 by 2 (since it’s a square root).
   1.8062 ÷ 2 = 0.9031
3. Step 3: Find the antilog of 0.9031. Antilog of 0.9031 (from antilog table) = 8.000.
   Characteristic = 0, so there is 0+1 = 1 digit before the decimal point.
   Therefore, Antilog (0.9031) = 8.000.

NOTE: When dealing with negative characteristics (e.g., for numbers less than 1), ensure the characteristic is handled correctly, often by adjusting it to a suitable integer before division, for instance, by adding and subtracting a multiple of the divisor.

Quick Revision Table

Area	Key Point
Revision of Laws of Indices	Indices (or exponents) represent the number of times a base number is multiplied by itself. For
Theory of Logarithm	A logarithm is the power to which a base must be raised to produce a given number. It is the inverse operation to exponentiation. If a x = N, then…
Laws of Logarithm	Logarithm laws are derived from the laws of indices and are used to simplify logarithmic expressions and calculations. The following are the main…
Using Logarithm Tables for Calculations	Logarithm tables are used to find the logarithm of numbers to base 10 (common logarithms) and to perform complex calculations. Every logarithm…

Teaching Methods/Instructional Techniques

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

Instructional Procedures

Step 1: Introduction and Recall

Time: 4 minutes

Teaching Skill: Set Induction

Teacher’s Activity: The teacher greets the students, checks attendance, and reviews the previous lesson briefly. The teacher then introduces the topic "Logarithms" by asking students how they would solve very large multiplication problems or find roots of complex numbers without a calculator, hinting at the usefulness of logarithms.

Students’ Activity: Students respond to greetings, answer questions about the previous lesson, and listen attentively to the introduction.

Learning Point: Prior logarithm knowledge.

Step 2: Revision of Laws of Indices and Theory of Logarithm

Time: 5 minutes

Teaching Skill: Explanation/Recall

Teacher’s Activity: The teacher guides students to recall the basic laws of indices, writing them on the board. The teacher then defines logarithm as the inverse of exponentiation and explains the relationship between index form (a x = N) and logarithm form (x = log a N), using simple examples.

Students’ Activity: Students recall and state laws of indices. They define logarithm and identify the relationship between index and logarithm forms.

Learning Point: Index and logarithm laws.

Step 3: Laws of Logarithm

Time: 5 minutes

Teaching Skill: Demonstration/Explanation

Teacher’s Activity: The teacher presents the three main laws of logarithm (product, quotient, and power laws) using the logarithm law chart. For each law, the teacher explains it with a simple example and guides students to verify it. The teacher emphasizes that these laws simplify calculations.

Students’ Activity: Students study the charts, deduce the laws, and verify them with simple exercises provided by the teacher.

Learning Point: Laws of logarithms.

Step 4: Using Logarithm Tables for Calculations

Time: 5 minutes

Teaching Skill: Demonstration/Guided Practice

Teacher’s Activity: The teacher explains the characteristic and mantissa of a logarithm. The teacher demonstrates how to find the logarithm of numbers (including those less than 1) and antilogarithms using the logarithm table. The teacher then guides students through worked examples of calculations (multiplication, division, powers, and roots) using the logarithm table, emphasizing clear steps.

Students’ Activity: Students pay attention, ask questions for clarification, and practice finding logarithms and antilogarithms. They follow the steps for solving calculation problems using the logarithm table.

Learning Point: Logarithm table calculations.

Step 5: Guided Practice

Time: 5 minutes

Teaching Skill: Guided Practice/Problem Solving

Teacher’s Activity: The teacher gives two additional problems drawn from logarithm and guides students to choose the correct method, show each stage of working and check their answers.

Students’ Activity: Students solve the guided problems, compare their working with the demonstrated method and correct any errors.

Learning Point: Logarithm guided practice.

Step 6: Application and Reinforcement

Time: 5 minutes

Teaching Skill: Individual Practice/Reinforcement

Teacher’s Activity: The teacher presents a different logarithm problem and asks students to identify the relevant rule or formula before solving it.

Students’ Activity: Students identify the appropriate method, solve the problem independently and check the reasonableness of the answer.

Learning Point: Logarithm practical application.

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 the relationship between indices and logarithms.
2. State the product law of logarithm with an example.
3. State the power law of logarithm with an example.
4. Explain how to determine the characteristic of log 0.0057.
5. Outline the steps to calculate 8.5 × 1.3 using a logarithm table.

Students’ Activity: Students answer orally and in writing, showing working or giving examples where the lesson requires it.

Learning Point: Logarithm skills assessed.

Step 8: Note-Taking

Time: 8 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides students to copy the essential Board Summary on logarithm into their notebooks, drawing attention to the key rules, examples, diagrams or distinctions already taught.

Students’ Activity: Students copy the essential notes carefully into their notebooks and correct any errors in their earlier work.

Learning Point: Logarithm notes recorded.

Step 9: Conclusion

Time: 3 minutes

Teaching Skill: Reinforcement/Summarization

Teacher’s Activity: The teacher summarizes the key learning points of the lesson, reiterating the importance of laws of indices, the definition of logarithms, the laws of logarithms, and the use of logarithm tables for calculations.

Students’ Activity: Students listen to the summary and copy down the assigned homework.

Learning Point: Logarithm concepts consolidated.

Continuous Assessment/Further Study

Type: Practice Exercise

Instruction: Complete the following tasks in your notebook. Show all mathematical working clearly.

1. Simplify \(\log_a(36x)-\log_a(6)\) as far as possible.
2. Express \(2\log_a 3+\log_a 5\) as a single logarithm.
3. Use a logarithm table to find \(23.6\times4.18\), showing the characteristic and mantissa used.
4. Use a logarithm table to evaluate \(\frac{86.4}{3.27}\), showing each stage.
5. State the relationship between the index form \(a^x=N\) and the logarithmic form.

Lesson Keywords

• Logarithm – The power to which a number (base) must be raised to get some other number.
• Index/Exponent – The power or exponent of a number.
• Base – The number that is repeatedly multiplied in exponentiation, or the number to which a logarithm is taken.
• Characteristic – The integral part of a common logarithm.
• Mantissa – The positive decimal part of a common logarithm.
• Antilogarithm – The number whose logarithm is given.

Differentiation

• Fast learners: Solve advanced logarithmic problems involving combined laws and multi-step calculations using logarithm tables independently.

• Average learners: Practice standard examples on multiplication, division, powers, and roots with minimal teacher support.

• Slow learners: Receive additional guidance on identifying characteristic and mantissa, applying basic laws of logarithms, and following step-by-step procedures using the logarithm table.

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