MATHEMATICS: FORM TWO: Topic 4 – LOGARITHMS

Tutorial on Standard Form

Standard Form is also called Scientific Notation. It is a way of writing a number into two parts. For example

• Thedigits(with the decimal point placed after the first digit) followed by
• X 10 to a powerthat puts a decimal point where it should be (i.e. it shows how many places to move the decimal point).

Numbers in Standard Form

Write numbers in standard form

Computations which Involved Multiplication and Division of Numbers Expressed in Standard Form

Perform computations which involved multiplication and division of numbers expressed in standard form

There are several laws of logarithms which help in evaluating them. These laws are valid for only positive real numbers. The laws are as follows:

Verify the laws of logarithms using the knowledge of exponents

Use the laws of logarithms to evaluate the following:

Change of base

This is a formula for change of base. For any positive a, b (a, b ≠0) we have

Given,

then find a number which is a common base to both 8 and 4

Solve.

Logarithms to Base 10

Use the laws of logarithms to evaluate the following:

Solutions

Most of the logarithmic tables are of base 10 (common logarithms). When we want to read a logarithm of a number from logarithmic table, we first check if the number is between 0 and 10 (but not 0 or 10) because the table consists only of logarithms of numbers between 0 and 1.

For example; what is the logarithm of 5.25 from the table. Our number is between 0 and 1. We look at the most left column and find where 52 is (we ignore the decimal point). Then slide your finger along this row to the right to find column of the next digit in our example is 5. Read the number where the row of 52 meets the column of 5. The logarithm of 5.25 is 0.7202.

If the number has 4 digits like 15.27, we do the following. First of all, checking our number we see that it is greater than 10. The number is between 10 and 100. And we know that the logarithm of 10 is 1 and logarithm of 100 is 2. So logarithm of 15.27 is between 1 and 2, normally less than 2 but greater than 1, hence 1.something. That something we need to find it in a logarithm table. Look at the most left column the row labeled 15, then, slide your finger to the right to find the column labeled 2. Read the number where the row of 15 meets the column of 2, the number is 0.1818. We are remaining with one digit which is 7. If your log table has a part with mean difference table, slide your finger over to the column in that table marked with the next digit of the number you are looking up, in our example it is 7. Slide over to row 15 and mean difference 7. The row of 15 meets mean difference column 7 at number 20. Add the two numbers obtained (the mean difference number is added to the last digits of our first number we obtained) i.e. 0.1818 + 20 = 1838. Now add characteristic which is 1 since 15.27 is between 10 and 100. We get 1 +0.1838 = 1.1838. ThereforeLog 15.27= 1.1838.

Note that if you are given a number with more than 4 digits, first round off the number to 4 digits and then go on with similar procedures as explained in examples above.

To find the number whose logarithm is known, we can call it ant-logarithm the same logarithmic table can be used. For example to find the number whose logarithm is 0.7597, look at the central part of the log table find the number (mantissa) 7597. This is in the intersection of the row labeled 57 and column 5. So the number is 575. But in order to get correct answer we have to consider characteristic of our logarithm which is 0. This means our number is between 0 and 10 because the numbers whose logarithms are 0.something are between 0 and 10. Hence, we need to place one decimal point from left to our number to make it be between 0 and 10. Therefore the number will be 5.75 i.e. log 5.75 = 0.7597, thus antilog 0.7597= 5.75.

How to find the ant-log

Step1: Understand the ant-log table. Use it when you have log of a number but not the number itself. the ant-log is also known as the inverse log.

Step 2: Write down the characteristic. This is the number before decimal point. If you are looking up the ant-log of 2.8699, the characteristic is 2. Remove it from the number you are looking up. But never forget it because it will be used later. So it is better if you write it somewhere.

Step 3: Find the row in the most left column that matches the first two numbers of the mantissa. Our mantissa is 8699. So run your finger down that column until you find .86.

Step 4: Slide your finger over to the column marked with the next digit of the mantissa. For 2.8699, slide your finger along the row marked .86 to find the intersection with column 9. This reads 7396. Write this down.

Step 5: If your ant-log table has a table of mean difference, slide your finger over to the column in that table marked with the next digit of the mantissa. Make sure to keep your finger in the same row. Considering our example, slide your finger over the to the last column in the table, column 9. The intersection of row .86 and mean difference column 9 is 15. Write it down.

Step 6: Add the two numbers obtained from the two previous steps. In our example, these are 7396 and 15. Adding them i.e. 7396 + 15 = 7411.

Step 7: Use characteristic to place decimal point. Our characteristic is 2, which means our answer is between 100 and 1000 because log 100 = 2 and log 1000 = 3. For the number 7411 to fall between 100 and 1000, the decimal point should be placed after 3 digits. So, the final answer is 741.1 therefore the ant-log of 2.8699 is 741.1.

Operations with Logarithms

Here we deal with all 4 operations which are addition, subtraction, multiplication and division. All operations are just as usual operations except division when we are given a negative characteristic. For example;

evaluate the following:

Find the product of 25.75 ×450.

Calculate using logarithms