Mathematics Lesson Plan: Logarithms (Week 8)

MATHEMATICS LESSON PLAN

WEEK: 8

TOPIC: LOGARITHMS

CLASS: Senior Secondary School 1 (SSS 1)

OBJECTIVES OF THE LESSON

By the end of this weekly module, students will be able to:

Define a logarithm and convert statements between Index Form and Logarithmic Form.
Identify the two distinct parts of a base-10 logarithm: the Characteristic and the Mantissa.
Determine the common logarithm of numbers greater than, equal to, and less than 1 using 4-figure tables.
Perform the inverse mathematical operation using Antilogarithm tables.
Apply the logarithmic laws of multiplication and division to compute complex numerical products and quotients.

1. THE FUNDAMENTAL CONCEPT OF LOGARITHMS

Definition: In general, the logarithm of a number y to a given base b is the power (or index) x to which that base must be raised to yield the number.

Mathematically: If b^x = y, then log_b(y) = x.

Therefore, the logarithm of a number to Base 10 is the specific power to which 10 must be raised to produce that number. For example, log₁₀(1000) = 3 because 10³ = 1000.

Visual Mapping: Index Form vs. Logarithmic Form

INDEX (EXPONENTIAL) FORM
b^x = y

⟺

LOGARITHMIC FORM
log_b(y) = x

Worked Examples: Conversions

Example 1: Express the following in Logarithmic Form

(a) 2^-6 = ¹/₆₄

➔ log₂(¹/₆₄) = -6

(b) 3⁵ = 243

➔ log₃(243) = 5

(c) 5³ = 125

➔ log₅(125) = 3

(d) 10⁴ = 10,000

➔ log₁₀(10000) = 4

Example 2: Express the following in Index Form

(a) log₂(¹/₈) = -3

➔ 2^-3 = ¹/₈

(b) log₁₀(¹/₁₀₀) = -2

➔ 10^-2 = ¹/₁₀₀

(c) log₄(64) = 3

➔ 4³ = 64

(d) log₅(625) = 4

➔ 5⁴ = 625

2. LOGARITHMS OF NUMBERS TO BASE 10

Every common logarithm consists of two distinct parts separated by a decimal point:

Logarithm Value = [ Integer Part: CHARACTERISTIC ] . [ Decimal Part: MANTISSA ]

Example: In log₁₀(3900) = 3.5911, the Characteristic is 3, and the Mantissa is .5911

Rules for Finding the Characteristic

For numbers ≥ 1: Count the number of digits to the left of the decimal point and subtract 1. (e.g., 51.38 has 2 digits before the decimal; 2 - 1 = 1).
For numbers < 1 (Decimals): Write the number in standard form (A × 10^-n). The characteristic is negative n, written in Bar Notation as n. (e.g., 0.009321 → 9.321 × 10^-3; Characteristic = 3).

Using the 4-Figure Logarithm Table (Finding the Mantissa)

To find the mantissa of 37, look down the left-hand margin for 37, and slide across to the column headed by 0:

x	0	1	2	...	7	8	9
35	5441	5453	5465	...	5527	5539	5551
36	5563	5575	5587	...	5647	5658	5670
37	5682	5694	5705	...	5763	5775	5786

Putting it together:

For 37: Characteristic is 1. Mantissa is .5682 → log₁₀(37) = 1.5682
For 3900: Characteristic is 3. Mantissa of 39 under 0 is .5911 → log₁₀(3900) = 3.5911

3. ANTILOGARITHMS

An Antilogarithm is the exact inverse of a logarithm. When finding an antilogarithm:

Look up only the fractional mantissa part inside the Antilogarithm table to get your raw sequence of digits.
Use the integer Characteristic to place the decimal point: Shift the decimal point (Characteristic + 1) places to the right.

Given Logarithm	Mantissa Lookup	Characteristic shift	Final Antilog Number
0.5682	.56 under 8, diff 2 → `3700`	0 + 1 = 1 digit	3.700
2.7547	.75 under 4, diff 7 → `5684`	2 + 1 = 3 digits	568.4
2.7652	.76 under 5, diff 2 → `5824`	Negative 2 → insert 1 zero	0.05824

4. MULTIPLICATION AND DIVISION USING TABLES

We use the standard Laws of Logarithms to turn multi-digit arithmetic into basic addition and subtraction:

Product Law: log(A × B) = log A + log B
Quotient Law: log(A ÷ B) = log A − log B

Worked Layouts

1. Evaluate: 4627 × 29.3

Number	Logarithm
4627	3.6653
× 29.3	+ 1.4669
Antilog: 135,600	5.1322

*Added the logs due to Product Law

2. Evaluate: 819.8 ÷ 3.905

Number	Logarithm
819.8	2.9137
÷ 3.905	− 0.5916
Antilog: 209.9	2.3221

*Subtracted the logs due to Quotient Law

3. Evaluate combined operation: (48.63 × 8.53) ÷ (15.39 × 3.52)

Operation Step	Number	Logarithm
Numerator	48.63 8.53	1.6869 + 0.9309
Numerator Sum (A)		2.6178
Denominator	15.39 3.52	1.1872 + 0.5465
Denominator Sum (B)		1.7337
Subtract: (A) − (B)		0.8841
Final Antilog Answer	7.658

CLASS EVALUATION

Find the logarithm of: (i) 0.009321 (ii) 0.5454
Find the antilogarithm of: (i) 3.3210 (ii) 1.8113 (iii) 1.5813 (iv) 0.2212
Use logarithm tables to evaluate: (36.12 × 750.9) ÷ (113.2 × 9.98)

READING ASSIGNMENT

New General Mathematics SSS1 — Read Page 21, complete Exercise 1h (Problems 1 – 3).

WEEKEND ASSIGNMENT

Part A: Multiple Choice Questions

Find the log of 802 to base 10 (use log tables):
(a) 2.9042 (b) 3.9040 (c) 8.020 (d) 1.9042
Find the number whose logarithm is 2.8321:
(a) 6719.2 (b) 679.4 (c) 0.4620 (d) 67.92
What is the integer characteristic of the log of 0.000352?:
(a) 4 (b) 3 (c) 4 (d) 3
Given that log₂(¹/₆₄) = m, what is m?:
(a) -5 (b) -4 (c) -6 (d) 3
Express in index form: log₁₀(10000) = 4:
(a) 10³ = 10000 (b) 10^-4 = 10000 (c) 10⁴ = 10000

Part B: Theory Questions

Evaluate using a logarithm table: (6.28 × 304) ÷ 981.
Express your final answer in the standard form A × 10ⁿ.
Use a logarithm table to calculate correct to 3 significant figures: (6354 × 6.243) ÷ (16.76 × 323)

🔒 CLICK TO REVEAL TEACHER'S ANSWER KEY

Class Evaluation Answers:

1(i) 3.9695 | 1(ii) 1.7367
2(i) 2094 | 2(ii) 64.76 | 2(iii) 0.3813 | 2(iv) 1.664
3. Numerator Log sum (4.4332) − Denominator Log sum (3.0530) = 1.3802 → Antilog = 23.99

Weekend Assignment (Part A - MCQ):

1. (a) 2.9042
2. (b) 679.4
3. (a) 4
4. (c) -6
5. (c) 10⁴ = 10000

Weekend Assignment (Part B - Theory Solutions):

Theory 1: Num Logs: 0.7980 + 2.4829 = 3.2809. Den Log: 2.9917.
Difference 3.2809 − 2.9917 = 0.2892. Antilog = 1.946 → Standard Form: 1.946 × 10⁰

Theory 2: Num Logs Sum = 4.5984. Den Logs Sum = 3.7335.
Difference = 0.8649. Antilog = 7.3265 → Correct to 3 s.f. = 7.33

Logarithms of Numbers to Base 10