**INTRODUCTION**

The logarithm is an important and diversified topic in mathematics. In this article, we will discuss the logarithm of numbers, teach you how to use the logarithm table, carry out logarithm examples with solutions and finally show you how to find the antilogarithm which is the converse of the logarithm.

The logarithm of numbers consists of two parts called the **Characteristics** and the** Mantissa**. A characteristic is a whole number that can either be positive, zero, or negative i.e. the character is an INTEGER. While the Mantissa is a DECIMAL i.e. fractional number and always positive. In the next few paragraphs, we will explore two ways of finding the characteristics of a number. And subsequently, teach you how to get the mantissa from the logarithm table.

The characteristics part obtained by two methods:

- By subtracting 1 from the number of digits in the
**integral**(i.e**23**.89) part of the number whose logarithm is sought; - By considering the standard form of the number. That is in which the index “n” is the character.

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**LOGARITHM OF NUMBERS CHARACTERISTICS**

Let us explain the two methods using the following examples. Consider the numbers

i. ii. iii. iv.

Using the first method above, that is Subtracting 1 from the number of digits in an integral part of the number, we have:

- , has digits in that integral numbers, so . The characteristics of the log of is ;
- , has digits in the integral number, so . The characteristics of the log of is ;
- , has digits in the integral number, so . The characteristics of the log of is ;
- , has digit in the integral part of the number, so . The characteristics of the log of is .

On the other hand, using the Standard form method, we will have the following:

- in standard form is . The characteristics of the log of is the index, which is 2;
- in standard form is . The characteristics of the log of is the index, which is 5;
- in standard form is . The characteristics of the log of is the index, which is 1;
- in standard form is . The characteristics of the log of is the index, which is 0.

So by whatever of the two methods above, we can get the characteristics of the logarithm of any number. I usually prefer the standard form method, this method makes it easy to identify negative characteristics easily. For example, the characteristics of log is which is the index of the standard form. And for the number , the characteristics is . The negative characteristics are normally written as bar, when solving questions on logarithm i.e. written as , written as etc.

**WATCH**: Standard Form: Keynotes for Solving Examination Questions.

**LOGARITHM OF NUMBERS MANTISSA**

The Mantissa part of the logarithm is read from the logarithm tables. Using the same examples, we will show you how to find the mantissa part of the logarithm.

For the number , we will look up for the first two digits in the column of the logarithm table. And follow the line across till we come under the third digit , here we get . Remember that the mantissa is always decimal, so this value will be .

Now combining the characteristics we got above with the mantissa part, we have the logarithm of to be .

Similarly, for, , since our table is a four-figure table only, this number becomes (i.e to four significant figures). So we look up along the column in the table and across under , we get and then the fourth digit , is the difference column. Across from under , in the difference column is . We ADD this to to get . Remember this is a decimal, i.e . Combining this with the characteristics above, we have the log of to be .

So what we simply did is: Log of under gives . Difference is . So we simply add to to get . Finally we included the characteristics of the number which is to get . **So you see, It’s easy!! See the table below for the results of the other examples given above.**

**LOGARITHM TABLE RESULTS**

The results of all the examples given above and more are tabulated in the table below:

Numbers | Subtract 1 from the digit in an integral part | Characteristic of Logarithm | Mantissa from the Log. Table | Add Difference | Total Mantissa | Logarithm |

369 | 3 – 1 = 2 | 2 | 36 under 9 = .5670 | None | .5670 | 2.5670 |

561728 | 6 – 1 = 5 | 5 | 56 under 1 = .7490 | Under 7 = 5 | .7495 | 5.7495 |

12 | 2 – 1 = 1 | 1 | 12 under 0 = .0792 | None | .0792 | 1.0792 |

2.136 | 1 – 1 = 0 | 0 | 21 under 3 = .3284 | Under 6 = 12 | .3296 | 0.3296 |

7 | 1 – 1 = 0 | 0 | 70 under 0 = .8451 | None | ..8451 | 0.8451 |

Logarithm makes complex calculations easier. These calculations are usually presented in tabular forms as we will see in some real examples at the end of this article. Note that Logarithm tables are usually calculated to 4 decimal places and that the Antilogarithm tables are used to get back the number required. This will be discussed in the next paragraph, but before we talk about that, let’s consider the logarithm of squares, square roots, cube, cube roots, etc.

**LOGARITHM WITH ROOTS AND POWERS**

**POWERS**: Whenever a logarithm question is raised to a power “n” i.e. , we can find the value of the logarithm using the logarithm table. After which, **multiply** the result from the table by the power which is “n” . So if the number is raised to power , we multiply the result from the logarithm table by , if raised to power , multiply the result from the logarithm table by and so on.

**ROOTS**: When we talk about roots, we mean, square roots, cube roots, fourth roots, etc. An example is the values , , etc. Generally, when finding the nth root of any number, the logarithm result of the number is **divided** by the root. So to find the cube root of a number, divide the result of the logarithm by 3. This also applies to all other roots.

In some questions, you might be required to find using a logarithm table the values of

i. and

ii.

The table below shows the solutions to the question above using the logarithm table.

So, when solving questions on logarithm involving roots and powers, for roots, **divide the logarithm result with the value of the root**. And for Power, **multiply the logarithm result with the “index” or power value**.

Download: Mathematics Textbooks with Past Questions and Answers

**ANTILOGARITHMS**

The Antilogarithm is the converse of the logarithm. All students should note these two points when finding the antilogarithms of a given number:

- The mantissa (i.e. the decimal part) of the given logarithm is looked for in the Antilogarithm table only;
- For the Characteristics, one is added to the characteristics part of the logarithm value to get the antilogarithm value. This value will determine where to fix the decimal place in the final result. The examples below will explain more.

Example: Find the number whose logarithm is i. ii.

: Here we are looking for the antilog of . From the antilogarithm table, look for (on the column), under . This will give you , then look under in the Difference column for the difference, which is . We will now **add** the difference to the main value, which is to get . Since the characteristics of the logarithm value is , we will add to it, this will determine the decimal point in the antilog result. So .

∴ the antilog of is , with the Characteristics, the required number is

: Look for under , this gives . Next, look under in the difference column, which gives . Finally, add to the main value, to get . Since the characteristics is , we add to it, this gives .

∴The antilog of is , with the characteristics, the required number is .

Click HERE to download a copy of the Logarithm and Antilogarithm part of the Four Figure Table.

**LOGARITHM EXAMPLES WITH SOLUTIONS**

**Sample 1**: Evaluate using a logarithm table.

Considering the Characteristics, . The logarithm of is

**Sample 2:** Evaluate .

Recall that the characteristic is . So the result of the above evaluation is .

The third sample is a video example of a complex question. Watch and learn from the way the question is solved.

**LOGARITHM VIDEO EXAMPLE**

For detailed quizzes on the logarithm of numbers, download the worksheet. Click HERE to download.

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