Graph of y = 10x, Finding Logarithms and Antilogarithms for SS 1

Class: SS 1
Term: First Term
Week: 8
Age: 15 years
Duration: 60 minutes
Subject: 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: Logarithms, Deducing from Indices and Standard
Topic: Logarithms: Graph Of Y = 10.X
Subject Matter: Graph of y = 10.x, Finding of logarithm and antilogarithm

Specific Objectives

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

Cognitive Domain

• Define a logarithm.
• State the relationship between indices and logarithms.
• State the basic laws of logarithms for multiplication, division, powers, and roots.

Affective Domain

• Appreciate the use of logarithms in simplifying complex calculations.
• Demonstrate patience and precision when plotting graphs and using logarithm tables.

Psychomotor Domain

• Plot the graph of (y = 10^x).
• Use digital tools or logarithm tables to find the logarithm of a given number.
• Use digital tools or logarithm tables to find the antilogarithm of a given number.
• Solve problems involving multiplication, division, power, and roots using logarithms.

Social Domain

• Collaborate effectively in groups to plot graphs and solve problems.

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
• New General Mathematics for Senior Secondary Schools 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
• Scientific calculators
• Video clip to find the Log and antilog of a given number
• Computer/Projector (for digital tools/video)

Rationale for the Lesson

This lesson is important as it introduces students to logarithms, a powerful mathematical tool used to simplify complex calculations involving multiplication, division, powers, and roots. Understanding logarithms provides a foundation for advanced mathematical concepts and has practical applications in various fields like science, engineering, and finance.

Prerequisite/Previous Knowledge

Students should have a good understanding of indices (exponents) and their laws, as well as basic graph plotting techniques for linear equations.

Lesson Content/Board Summary

Logarithms: Graph Of Y = 10.X

Introduction to Logarithms

A logarithm expresses the exponent to which a base must be raised to obtain a number. It is the inverse operation of exponentiation (indices).

If (a^x = N), then (x) is the logarithm of (N) to the base (a). This is written as (x = log_a N).

1. The number (a) is called the base.
2. The number (x) is the logarithm.
3. The number (N) is the argument.

For example, since (10^2 = 100), then (log_{10} 100 = 2).

Common logarithms (base 10) are often written without the base, e.g., (log 100 = 2).

Graph of (y = 10^x)

The graph of (y = 10^x) is an exponential curve. It shows the relationship between a number and its logarithm to base 10. We can plot this graph by choosing various values for (x) and calculating the corresponding values for (y).

Steps to plot the graph of (y = 10^x):

1. Create a table of values for (x) and (y). Choose a range of (x) values (e.g., from -2 to 2).
2. Calculate (y = 10^x) for each chosen (x).
3. Choose an appropriate scale for both the x-axis and y-axis.
4. Plot the points ((x, y)) on the graph paper.
5. Draw a smooth curve through the plotted points.

Example Table of Values for (y = 10^x):

(x)	(y = 10^x)
-2	(10^{-2} = 0.01)
-1	(10^{-1} = 0.1)
0	(10^0 = 1)
1	(10^1 = 10)
2	(10^2 = 100)

The graph will pass through ((0, 1)) and will always be above the x-axis, approaching it as (x) becomes very negative.

Finding Logarithms of Numbers Greater Than 1

To find the logarithm of a number (e.g., (log_{10} N)), we determine two parts:

1. Characteristic: This is the integer part of the logarithm. It is determined by the position of the decimal point in the number (N). If (N ge 1), the characteristic is one less than the number of digits before the decimal point.
2. Mantissa: This is the decimal part of the logarithm. It is found using logarithm tables or a scientific calculator. The mantissa is always positive.

Example: Find (log_{10} 345.6)

1. Characteristic: There are 3 digits before the decimal point (3, 4, 5). So, the characteristic is (3 – 1 = 2).
2. Mantissa: Look up the mantissa for 3456 in a logarithm table. (e.g., in a 4-figure table, it might be .5386).

So, (log_{10} 345.6 = 2.5386).

Finding Antilogarithms

Antilogarithm is the inverse process of finding a logarithm. If (log_{10} N = x), then (N = text{antilog } x = 10^x).

To find the antilogarithm of a number (e.g., (text{antilog } x)), we use the characteristic and mantissa:

1. Mantissa: Use the mantissa (decimal part) to find the sequence of digits from the antilogarithm table or calculator.
2. Characteristic: Use the characteristic (integer part) to place the decimal point. If the characteristic is (C), the number of digits before the decimal point will be (C + 1).

Example: Find (text{antilog } 2.5386)

1. Mantissa: Look up .5386 in the antilogarithm table. This corresponds to the sequence of digits 3456.
2. Characteristic: The characteristic is 2. So, the number of digits before the decimal point is (2 + 1 = 3).

Therefore, (text{antilog } 2.5386 = 345.6).

Laws of Logarithms

These laws are used to simplify calculations involving logarithms.

1. Multiplication Law: The logarithm of a product is the sum of the logarithms of the individual numbers.

   (log_b (MN) = log_b M + log_b N)

2. Division Law: The logarithm of a quotient is the difference between the logarithms of the numerator and the denominator.

   (log_b left(frac{M}{N}right) = log_b M – log_b N)

3. Power Law: The logarithm of a number raised to a power is the power multiplied by the logarithm of the number.

   (log_b (M^p) = p log_b M)

4. Root Law: The logarithm of the (p)-th root of a number is the logarithm of the number divided by (p). This is a special case of the power law since (sqrt[p]{M} = M^{1/p}).

   (log_b (sqrt[p]{M}) = frac{1}{p} log_b M)

Worked Examples

Example 1: Multiplication

Question: Use logarithms to evaluate (25.3 times 4.12).

Solution:

Let (P = 25.3 times 4.12)

Step 1: Take the logarithm of both sides.

(log P = log (25.3 times 4.12))

Step 2: Apply the multiplication law.

(log P = log 25.3 + log 4.12)

Step 3: Find the logarithms of individual numbers.

(log 25.3 = 1.4031) (Characteristic = 1, Mantissa from table for 2530)

(log 4.12 = 0.6149) (Characteristic = 0, Mantissa from table for 4120)

Step 4: Add the logarithms.

(log P = 1.4031 + 0.6149 = 2.0180)

Step 5: Find the antilogarithm of the result.

(P = text{antilog } 2.0180)

Mantissa .0180 corresponds to 1042 (from antilog table).

Characteristic 2 means (2+1=3) digits before the decimal point.

(P = 104.2)

Answer: (25.3 times 4.12 = 104.2)

Example 2: Division

Question: Use logarithms to evaluate (frac{789.2}{12.5}).

Solution:

Let (Q = frac{789.2}{12.5})

Step 1: Take the logarithm of both sides.

(log Q = log left(frac{789.2}{12.5}right))

Step 2: Apply the division law.

(log Q = log 789.2 – log 12.5)

Step 3: Find the logarithms of individual numbers.

(log 789.2 = 2.8972)

(log 12.5 = 1.0969)

Step 4: Subtract the logarithms.

(log Q = 2.8972 – 1.0969 = 1.8003)

Step 5: Find the antilogarithm of the result.

(Q = text{antilog } 1.8003)

Mantissa .8003 corresponds to 6314.

Characteristic 1 means (1+1=2) digits before the decimal point.

(Q = 63.14)

Answer: (frac{789.2}{12.5} = 63.14)

Example 3: Power

Question: Use logarithms to evaluate ((3.45)^3).

Solution:

Let (R = (3.45)^3)

Step 1: Take the logarithm of both sides.

(log R = log (3.45)^3)

Step 2: Apply the power law.

(log R = 3 times log 3.45)

Step 3: Find the logarithm of the number.

(log 3.45 = 0.5378)

Step 4: Multiply the logarithm by the power.

(log R = 3 times 0.5378 = 1.6134)

Step 5: Find the antilogarithm of the result.

(R = text{antilog } 1.6134)

Mantissa .6134 corresponds to 4106.

Characteristic 1 means (1+1=2) digits before the decimal point.

(R = 41.06)

Answer: ((3.45)^3 = 41.06)

Example 4: Root

Question: Use logarithms to evaluate (sqrt[3]{87.6}).

Solution:

Let (S = sqrt[3]{87.6} = (87.6)^{1/3})

Step 1: Take the logarithm of both sides.

(log S = log (87.6)^{1/3})

Step 2: Apply the power (root) law.

(log S = frac{1}{3} times log 87.6)

Step 3: Find the logarithm of the number.

(log 87.6 = 1.9425)

Step 4: Divide the logarithm by the root index.

(log S = frac{1.9425}{3} = 0.6475)

Step 5: Find the antilogarithm of the result.

(S = text{antilog } 0.6475)

Mantissa .6475 corresponds to 4441.

Characteristic 0 means (0+1=1) digit before the decimal point.

(S = 4.441)

Answer: (sqrt[3]{87.6} = 4.441)

Teaching Methods/Instructional Techniques

Discussion, Demonstration, Guided Practice, Question and Answer, Explanation, Problem Solving, Pair Work, Group Work, Individual Practice.

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Recalling/Connecting

Teacher’s Activity: The teacher reviews the concept of indices and asks students to express numbers in index form (e.g., (100 = 10^2), (1000 = 10^3)). The teacher then introduces logarithms as the inverse of indices.

Pupils’ Activity: Pupils respond to questions on indices and listen attentively to the introduction of logarithms.

Learning Point: Indices and logarithms connection

Step 2: Definition and Relationship to Indices

Time: 10 minutes

Teaching Skill: Explanation/Illustration

Teacher’s Activity: The teacher defines a logarithm and explains its relationship to index notation using examples (e.g., (10^2 = 100 implies log_{10} 100 = 2)). The teacher also introduces the concept of base 10 logarithms.

Pupils’ Activity: Pupils listen, ask questions for clarity, and write down the definition and examples.

Learning Point: Meaning of logarithm

Step 3: Plotting the Graph of (y = 10^x)

Time: 15 minutes

Teaching Skill: Demonstration/Guided Practice

Teacher’s Activity: The teacher guides students to brainstorm in groups to create a table of values for (y = 10^x) for (x) from -2 to 2. The teacher then demonstrates how to plot these points on a graph board, explaining the choice of scale and drawing a smooth curve. Students are encouraged to plot in their graph books.

Pupils’ Activity: Students work in groups to generate table values, then individually plot the graph of (y = 10^x) in their graph books.

Learning Point: Graphing exponential functions

Step 4: Finding Logarithms

Time: 5 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher explains how to find the logarithm of a number greater than 1, focusing on characteristic and mantissa. The teacher demonstrates using a logarithm table or a scientific calculator/digital tool to find logarithms of various numbers.

Pupils’ Activity: Pupils observe the demonstration and practice finding logarithms using their calculators or tables.

Learning Point: Finding number logarithms

Step 5: Finding Antilogarithms

Time: 5 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher explains the process of finding antilogarithms, which is the reverse of finding logarithms. The teacher demonstrates using antilogarithm tables or a scientific calculator/digital tool.

Pupils’ Activity: Pupils follow the demonstration and practice finding antilogarithms.

Learning Point: Finding number antilogarithms

Step 6: Laws of Logarithms and Problem Solving

Time: 10 minutes

Teaching Skill: Explanation/Problem Solving

Teacher’s Activity: The teacher introduces the laws of logarithms (multiplication, division, power, and root) and works through the examples provided in the Board Summary, showing step-by-step calculations using logarithms.

Pupils’ Activity: Pupils pay attention to the explanation, ask questions, and copy the worked examples.

Learning Point: Applying logarithm laws

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 a logarithm.
2. Express (10^3 = 1000) in logarithmic form.
3. Find (log 56.7) using a calculator or table.
4. Find (text{antilog } 1.7536).

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Logarithm concept understanding

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, including definitions, graph plotting steps, and worked examples, into their notebooks.

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

Learning Point: Recording lesson information

Step 9: Conclusion

Time: 5 minutes

Teaching Skill: Summarising

Teacher’s Activity: The teacher summarises the key points of the lesson, reiterating the connection between indices and logarithms, the process of finding logs and antilogs, and the importance of the laws of logarithms in simplifying calculations. The teacher encourages students to practice regularly.

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

Learning Point: Logarithm concept consolidation

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following problems using logarithm tables or a scientific calculator. Show all your working.

1. Evaluate (12.8 times 3.5).
2. Evaluate (frac{98.7}{2.34}).
3. Evaluate ((5.14)^4).
4. Evaluate (sqrt{789}).
5. Plot the graph of (y = 10^x) for (x) values from -1.5 to 1.5, using intervals of 0.5.

Lesson Keywords

• Logarithm – The exponent to which a base must be raised to produce a given number.
• Base – The number that is raised to a power in an exponential expression or the number whose power is the logarithm.
• Characteristic – The integer part of a logarithm.
• Mantissa – The positive decimal part of a logarithm.
• Antilogarithm – The number corresponding to a given logarithm; the inverse of a logarithm.
• Indices – Powers or exponents.
• Exponential curve – The shape of the graph of an exponential function like (y = 10^x).

Differentiation

For struggling learners: Provide simpler numbers for practice, offer one-on-one guidance during graph plotting, and allow the use of calculators for basic log/antilog finding before moving to complex problems. Provide pre-filled tables for graph plotting.

For advanced learners: Challenge them with more complex problems involving combinations of the laws of logarithms or introduce logarithms to other bases (e.g., natural logarithms). Encourage them to explore real-world applications of logarithms.

Suggested Lesson Videos

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