Logarithms

Subject :

Mathematics

Topic :

Logarithms

Class :

SS 1

Term :

First Term

Week :

Week 6

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Previous Knowledge :

The pupils have previous knowledge of

Standard form and Indices

Behavioural Objectives: At the end of the lesson, the students should be able to

define logarithm
deduce a relationship between logarithm and indices

use the graph of y=10x for multiplication and division
find the logarithm and antilog of numbers greater than one

Content:

TOPIC: LOGARITHMS CONTENT

Deducing logarithm from indices and standard form
Definition of Logarithms
Antilogarithms

The graph of y = 10^x

There is a close link between indices and logarithms.

Similarly, “log base e of 2” can be written as “e to the power of 1/2”, since 2 = e1/2.

One of the most useful properties of logarithms is that they can be used to solve equations in which the unknown quantity is an exponent. For instance, suppose we want to find out if a number raised to the power of 3 equals 27. We can use logs to solve this equation.

3x = 27

Take the log from both sides:

log 3x = log 27

Now we can use the fact that logs obey the same laws as indices to simplify this equation:

x log 3 = log 27.

Divide both sides by log 3:

x = (log 27) / (log 3).

Now we just need to find the value of (log 27) / (log 3). This can be done using a calculator or by looking up the values in a table of logarithms. We find that

x = 3

So the solution to the equation 3x = 27 is x = 3.

This method can be used to solve any equation in which the unknown quantity is an exponent. All we need to do is take the log of both sides of the equation and then use the laws of indices to simplify it. Finally, we just need to find the value of the expression on the left hand side using a calculator.

100 = 10². This can be written in logarithmic notation as log₁₀100 = 2. Similarly 8 = 2³ and it can be written as log₂8 = 3.

In general, N = b^x in logarithmic notation is Log_bN = x.

We say the logarithms of N in base b is x. When the base is ten, the logarithms is known as common logarithms.

The logarithms of a number N in base b is the power to which b must be raised to get N.

Evaluation.

Re-write using logarithmic notation (i) 1000 = 10³ (ii)0.01 = 10^-2 (iii)= 16 (iv) = 2^-3 Change the following to index form

(i) Log₄16 = 2 (ii) log₃ () = -3

The logarithm of a number has two parts and integer (whole number) then the decimal point. The integral part is called the characteristics and the decimal part is called mantissa.

To find the logarithms of 27.5 form the table, express the number in the standard form as 27.5

= 2.75 x 10¹. The power of ten in this standard form is the characteristics of Log 27.5. The decimal part is called mantissa.

Remember a number is in the standard form if written as A x 10ⁿ where A is a number such that 1 ≤ A < 10 and n is an integer.

27.5 = 2.75 x 10^1, 27.5 when written in the standard form, the power of ten is 1. Hence the characteristic of Log 27.5 is 1. The mantissa can be read from 4-figure table. This 4-figure table is at the back of your New General Mathematics textbook.

Below is a row from the 4-figure table

Differences

431

433

434

436

437

439

440

442

444

445

To check for Log27.5, look for the first two digits i.e 27 in the first column.

Now look across that row of 27 and stop at the column with 5 at the top. This gives the figure 4393.

Hence Log27.5 = 1.4393

To find Log275.2, 2.752 x 10²

The power of 10 is the standard form of the number is 2. Thus, the characteristic is 2. Log275.2

= 2. ‘Something’

For the mantissa, find the figure along the row of 27 with middle column under 5 as before (4393). Now find the number in the differences column headed. This number is 3. Add 3 to 4393 to get 4396. Thus Log275.2 = 2.4396.

Evaluation

Use table to find

Log 1
Log 71

Log 238

LESSON 2

If the logarithm of a number is given, one can determine the number for the antilogarithm table.

Example: Find the antilogarithms of the following (a) 0.5670 (b)2.9504 Solution

The first two digits after the decimal point i.e .56 is sought for in the extreme left column of the antilogarithm table then look across towards right till you, get to the column with heading 9. (Read 56 under 9) there you will see 3707. Since the integral part of 0.5690 is 0, it means if the antilog (3707) is written in the standard form the power of 10 is

i.e antilog of 0.5690 = 3.707 x 10⁰

= 3.707

The antilog of 9504 is found by checking the decimal part .9504 in the table.
Along the row beginning with 0.95 look forward right and pick the number in the column with 0 at the top. This gives the figures 8913 proceed further to the difference column with heading 4 to get This 8 is added to 8913 to get 8921. The integral part of the initial number (2.9504) is

This shows that there are three digits before the decimal point in the antilog of 2.9504 so

antilog 2.9504 = 892.1.

LESSON 3

The graph of y = 10^x can be used to find antilogarithm (and logarithm). Below is the table of values.

X	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
Y = 10^x	1	1.3	1.6	2.0	2.5	3.2	4.0	5.0	6.3	7.9	10

For example the broken line shows that the antilog 0.5 is approximately 3.2 or inversely that Log 3.2 ~0.5

Evaluation

Find the logarithms of
- (i) 32.7
- (ii) 61.02
- (iii) 3.247
Use antilog tables to find the numbers whose logarithms are
- (1) (1.82)
- (ii)2.0813
- (iii) 2108

Natural Log Rules

There are a large number of logarithmic rules. However, the major ones are listed below:

Logarithm product rule: logarithm of the multiplication of x and y is the sum of the logarithm of x and logarithm of y. [log_b(x ∙ y) = log_b(x)+ log_b(y)]

For example: log₁₀(4 ∙ 8) = log₁₀(4) + log₁₀(8)

Logarithm quotient rule: logarithm of the division of x and y provides us with the difference of logarithm of x and logarithm of y. [log_b(x / y) = log_b(x)- log_b(y)]

For example: log₁₀(4 / 8) = log₁₀(4) – log₁₀(8)

Logarithm power rule: logarithm of x raised to the power of y gives us y times the logarithm of x. [log_b(x ^y) = y ∙ log_b(x)]

For example: log₁₀(3⁸) = 8∙ log₁₀(3)

Logarithm base switch rule: logarithm of x to the base y is equal to 1 divided by logarithm of y to the base x. [log_b(x) = 1 / log_c(y)]

For example: log₃(4) = 1 / log₄(3)

Logarithm base change rule: logarithm of x to the base y is equal to the logarithm of x to the base z divided by the logarithm of y to the base z minus logarithm of x. [log_y(x) = log_z(x) / log_c(b) Logarithm – log(x)]

For example: log₄(3) = log₅(3) / log₅(4) Logarithm – log(3)

Logarithms II

LOGARITHMS

CONTENT:

Use Of Logarithm Table And Antilogarithm Table In Calculations Involving

Multiplication

Division
Powers
Roots

To multiply two numbers, look up each number in the logarithm table and add the logs together. Then, find this sum in the antilogarithm table to get the product

For example, to calculate 72 9 = 648, we would look up 7 and 2 in the logarithm table

Add the logs together to get 2.879 (which rounds to 2.88 in the antilogarithm table)

Then look up 2.88 in the antilogarithm table to find that 648 is the product

2. To divide two numbers, look up each number in the logarithm table and subtract the logs. Then, find this difference in the antilogarithm table to get the quotient

For example, to calculate 72 / 9 = 8, we would look up 7 and 2 in the logarithm table

Subtract the logs to get 2.879 (which rounds to 2.88 in the antilogarithm table)

Then look up 2.88 in the antilogarithm table to find that 8 is the quotient

3. To calculate a power, look up the base in the logarithm table and multiply by the exponent. Then, find this product in the antilogarithm table

For example, to calculate 72 = 4, we would look up 7 in the logarithm table and multiply by 2

This gives us 2.879 (which rounds to 2.88 in the antilogarithm table)

Then look up 2.88 in the antilogarithm table to find that 4 is the answer

4. To calculate a root, look up the number whose root you want to take in the logarithm table and divide by the exponent. Then, find this quotient in the antilogarithm table

For example, to calculate 72 = 9, we would look up 7 in the logarithm table and divide by 2

This gives us 2.879 (which rounds to 2.88 in the antilogarithm table)

Then look up 2.88 in the antilogarithm table to find that 9 is the answer

5. You can also use the logarithm table and antilogarithm table to calculate exponents

For example, to calculate e2 = 7.389, we would look up 2 in the logarithm table

This gives us 0.693 (which rounds to 0.69 in the antilogarithm table)

Then look up 0.69 in the antilogarithm table to find that 7.389 is the answer

In summary, the logarithm table and antilogarithm table can be used to simplify calculations involving multiplication, division, powers, and roots. These tables are also useful for calculating exponents.

Multiplication of Numbers using Logarithm Tables

Multiplication is one of the four basic operations in mathematics, along with addition, subtraction, and division. Although it might seem simple to multiply two numbers, there are actually a number of different ways to do so, each with its own advantages and disadvantages. One method is to use logarithm tables.

Logarithm tables are tables of numbers that have been calculated to logarithmic form. They can be used to simplify multiplication by allowing you to add the logarithms of the numbers you’re multiplying instead of multiplying the numbers themselves. For example, if you wanted to multiply 2 times 3, you could look up the logarithm of 2 in the table and add it to the logarithm of 3. This would give you the logarithm of 6, which you could then convert back into a regular number to get your answer.

Logarithm tables are most often used by people who need to do a lot of multiplication or who are working with very large numbers. They can be found in mathematical handbooks and are sometimes used by engineers and scientists.

1. To multiply 2 times 3, look up the logarithm of 2 in the table and add it to the logarithm of 3. This gives you the logarithm of 6, which you can convert back into a regular number to get your answer.

2. To multiply 3 times 4, look up the logarithm of 3 in the table and add it to the logarithm of 4. This gives you the logarithm of 12, which you can convert back into a regular number to get your answer.

3. To multiply 4 times 5, look up the logarithm of 4 in the table and add it to the logarithm of 5. This gives you the logarithm of 20, which you can convert back into a regular number to get your answer.

4. To multiply 5 times 6, look up the logarithm of 5 in the table and add it to the logarithm of 6. This gives you the logarithm of 30, which you can convert back into a regular number to get your answer.

5. To multiply 6 times 7, look up the logarithm of 6 in the table and add it to the logarithm of 7. This gives you the logarithm of 42, which you can convert back into a regular number to get your answer.

LOGARITHMS

CONTENT: Use Of Logarithm Table And Antilogarithm Table In Calculation Involving

Multiplication
Division

Powers

Roots
Application of logarithm in capital market and other real life problems

PERIOD 1

Logarithm and antilogarithm tables are used to perform some arithmetic basic operations namely: multiplication and division. Also, we use logarithm in calculations involving powers and roots.

The basic principles of calculation using logarithm depends strictly on the laws on indices. Recall from week 7 that.

Log MN = Log M + Log N
Log = Log M – Log N

Hence, we conclude that in logarithm;

When numbers are multiplied, we add their logarithms
When two numbers are dividing, we subtract their

SUB- TOPIC 1

Multiplication of numbers using logarithm tables

Example 1

Evaluate 92.63 x 2.914 Solution

Therefore 92.63 x 2.914 = 269.9

Example 2

Evaluate 34.83 x 5.427 Solution

Therefore 34.83 x 5.427= 189.1

EVALUATION

Evaluate the following 1. 6.26 x 23.83

409.1 x 3.932
8.31 x 22.45 x 19.64
431.2 x 21.35

SUB-TOPIC 2 – DIVISION OF NUMBERS USING LOGARITHM

Example 1

Evaluate 357.2 ÷ 87.23

Solution1

Therefore 357.2 x 87.23 = 4.096

Example 2

Use a logarithm table to evaluate 75.26 ÷ 2.581

Solution Number	Standard Form	Log	Operation
75.26	7.526 x 10¹	1.8765	Subtraction
2.581	2.581 x 10⁰	0.4118
		1.4647

Log Anti-Log Standard form Number

1.4647 2916 2.916 x 10¹ 29.16

Therefore 75.26 ÷ 2.581 = 29.16

Evaluation

Use table to evaluate the following

(1) 53.81 ÷ 16.25 (2)632.4 ÷ 34.25 (3) 63.75 ÷ 8.946 (4) 875.2 ÷ 35.81

SUB-TOPIC 3

Calculation Of Powers Using Logarithm

Study these examples.

At times, calculations can involve powers and roots. From the laws of logarithm, we have

Log Mⁿ = nLogM
Log M^1/n = Log M = Log
Log M^x/n = Log M =

Example 1.

Evaluate the following (53.75)³

Solution

Therefore 53.75³ = 155300

Example 2: 64.59²

Solution

Therefore 64.59²= 4173

Evaluation

Evaluate the following 1. 5.632⁴

19.18³
(67.9/5.23)³
x 92.6

SUB-TOPIC 4

Calculation of roots using Logarithms and Anti-Logarithms

Use tables of Logarithms and antilog to calculate Solution

Therefore = 1.939

Example 2: use table to find Solution

Hint: Workout 218 ÷ 3.12 before taking the cube root.

Number	Standard Form		Log	Operation
218	(2.18 x 10²)	2.3385		Subtraction
3.12	(3.12 x 10⁰)	0.4942
1.8		443 ÷ 3 Divisi 0.6148		on

Therefore = 4.119

Example 3

Evaluate 63.75² – 21.39²

We can use difference of two squares

i.e A² – B² = (A+B)(A-B)

63.75² – 21.39² = (63.75+21.39)(63.75-21.39)

= (85.14)(42.36)

= 85.14 X 42.36

Number	Log	operation
85.14	(1.9301)	Addition
42.36	(1.6260)
3606	3.5561
…. 3606
EVALUATION

Evaluate the following

(39.65² – 7.43²)^1/2
84.35² – 36.95²
64.74² – 55.262
94.68² – 43.25²
25.14² – 7.52²

Presentation

The topic is presented step by step

Step 1:

The subject teacher revises the previous topics

Step 2.

He or she introduces the new topic.

Step 3:

The subject teacher allows the pupils to give their own examples and he corrects them when the need arises.

Conclusion:

The subject teacher wraps up or concludes the lesson by giving out a short note to summarize the topic that he or she has just taught.

The subject teacher also goes round to make sure that the notes are well copied or well written by the pupils.

He or she makes the necessary corrections when and where the needs arise.

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Tags:First Term, MATHEMATICS

Mathematics

Logarithms

Standard form and Indices

Content:

Evaluation.

Evaluation

LESSON 2

LESSON 3

Evaluation

Natural Log Rules

PERIOD 1

SUB- TOPIC 1

Example 1

Example 2

EVALUATION

SUB-TOPIC 2 – DIVISION OF NUMBERS USING LOGARITHM

Solution1

Example 2

Evaluation

SUB-TOPIC 3

Example 1.

Solution

Evaluation

SUB-TOPIC 4

Example 3

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