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 \dpi{120} a\times 10^{n{\color{DarkOrange} }} in which the index “n” is the character.

RECOMMENDED: Surds in Mathematics


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

i. \dpi{120} 369       ii. \dpi{120} 561728       iii. \dpi{120} 12         iv. \dpi{120} 2.136

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

  1. \dpi{120} 369, has \dpi{120} 3 digits in that integral numbers, so 3-1=2. The characteristics of the log of \dpi{120} 369 is \dpi{120} 2 ;
  2. \dpi{120} 561728, has \dpi{120} 6 digits in the integral number, so 6-1=5. The characteristics of the log of \dpi{120} 561728 is \dpi{120} 5 ;
  3. \dpi{120} 12 , has \dpi{120} 2 digits in the integral number, so 2-1=1. The characteristics of the log of \dpi{120} 12 is \dpi{120} 1 ;
  4. \dpi{120} 2.136 , has \dpi{120} 1 digit in the integral part of the number, so 1-1=0. The characteristics of the log of \dpi{120} 2.136 is \dpi{120} 0.

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

  1. \dpi{120} 369 in standard form is 3.69\times 10^{2}. The characteristics of the log of \dpi{120} 369 is the index, which is 2;
  2. \dpi{120} 561728 in standard form is 5.61728\times 10^{5}. The characteristics of the log of \dpi{120} 561728 is the index, which is 5;
  3. \dpi{120} 12 in standard form is 1.2\times 10^{1}. The characteristics of the log of \dpi{120} 12 is the index, which is 1;
  4. \dpi{120} 2.136 in standard form is 2.136\times 10^{0}. The characteristics of the log of \dpi{120} 2.136 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, \dpi{120} 0.00264 = 2.64\times 10^{-3} the characteristics of log \dpi{120} 0.00264 is \dpi{120} -3 which is the index of the standard form. And for the number \dpi{120} 0.264=2.64\times 10^{-1} , the characteristics is \dpi{120} -1. The negative characteristics are normally written as bar, when solving questions on logarithm i.e. \dpi{120} -1 written as  \dpi{120} \overline{1} , \dpi{120} -2 written as \dpi{120} \overline{2} etc.

WATCH: Standard Form: Keynotes for Solving Examination Questions.


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 \dpi{120} 369, we will look up for the first two digits \dpi{120} 36 in the x column of the logarithm table. And follow the line across till we come under the third digit \dpi{120} 9, here we get \dpi{120} 5670. Remember that the mantissa is always decimal, so this value will be \dpi{120} {\color{Orange} .5670}.

Now combining the characteristics we got above with the mantissa part, we have the logarithm of \dpi{120} 369 to be \dpi{120} {\color{Orange} 2.5670}.

Similarly, for, \dpi{120} 561728, since our table is a four-figure table only, this number becomes \dpi{120} 561700 (i.e to four significant figures). So we look up \dpi{120} 56 along the \dpi{120} x column in the table and across under \dpi{120} 1, we get \dpi{120} 7490 and then the fourth digit \dpi{120} 7 , is the difference column. Across from \dpi{120} 56 under \dpi{120} 1 , in the difference column is \dpi{120} 5 . We ADD this to \dpi{120} 7490 to get \dpi{120} 7495. Remember this is a decimal, i.e \dpi{120} {\color{Orange} .7495}. Combining this with the characteristics \dpi{120} 5 above, we have the log of \dpi{120} 561728 to be \dpi{120} {\color{Orange} 5.7495}.

So what we simply did is: Log of \dpi{120} 56 under \dpi{120} 1 gives \dpi{120} 7490. Difference \dpi{120} 7 is \dpi{120} 5. So we simply add \dpi{120} 5 to \dpi{120} 7490 to get \dpi{120} 7495. Finally we included the characteristics of the number \dpi{120} 561728 which is \dpi{120} 5 to get \dpi{120} 5.7495. So you see, It’s easy!! See the table below for the results of the other examples given above.


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

NumbersSubtract 1 from
the digit in an
integral part
of Logarithm
from the
Log. Table
Add DifferenceTotal MantissaLogarithm
369 3 – 1 = 2 236 under 9 = .5670 None .5670 2.5670
561728 6 – 1 = 5 556 under 1 = .7490Under 7 = 5 .7495 5.7495
12 2 – 1 = 1 112 under 0 = .0792 None .0792 1.0792
2.136 1 – 1 = 0 021 under 3 = .3284 Under 6 = 12 .3296 0.3296
7 1 – 1 = 0 070 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.


POWERS: Whenever a logarithm question is raised to a power “n” i.e. \dpi{120} 2364^{n}, we can find the value of the logarithm \dpi{120} 2364 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 \dpi{120} 2 , we multiply the result from the logarithm  table by \dpi{120} 2 , if raised to power \dpi{120} 3, multiply the result from the logarithm table by \dpi{120} 3 and so on.

ROOTS: When we talk about roots, we mean, square roots, cube roots, fourth roots, etc. An example is the values \dpi{120} \sqrt[2]{2365} , \dpi{120} \sqrt[5]{7653} , \dpi{120} \sqrt[4]{9860} 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. \dpi{120} \sqrt[4]{190.8}  and

ii. \dpi{120} (24.7)^{2}

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

logarithm with squares and square root tablestable

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


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. \dpi{120} 2.6992   ii. \dpi{120} 0.4771

\dpi{120} 2.6992 : Here we are looking for the antilog of \dpi{120} 2.6992. From the antilogarithm table, look for \dpi{120} .69 (on the \dpi{120} x column), under \dpi{120} 9. This will give you \dpi{120} 5000 , then look under \dpi{120} 2 in the Difference column for the difference, which is \dpi{120} 2. We will now add the difference to the main value, which is \dpi{120} 5000 to get \dpi{120} 5002. Since the characteristics of the logarithm value is \dpi{120} 2 , we will add \dpi{120} 1 to it, this will determine the decimal point in the antilog result. So \dpi{120} 2+1=3.

∴ the antilog of \dpi{120} 2.6992 is \dpi{120} 5002 , with the Characteristics, the required number is \dpi{120} 500.2 

\dpi{120} 0.4771 : Look for \dpi{120} .47 under \dpi{120} 7, this gives \dpi{120} 2999. Next, look under \dpi{120} 1 in the difference column, which gives \dpi{120} 1. Finally, add \dpi{120} 1 to the main value, \dpi{120} 2999 to get \dpi{120} 3000. Since the characteristics is \dpi{120} 0, we add \dpi{120} 1 to it, this gives \dpi{120} 0+1=1.

∴The antilog of \dpi{120} 0.4771 is \dpi{120} 3000 , with the characteristics, the required number is \dpi{120} 3.000.

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


Sample 1: Evaluate \dpi{120} 92.63 \times 2.914 using a logarithm table.

Logarithm of numbers

Considering the Characteristics, \dpi{120} 2+1=3 . The logarithm of \dpi{120} 96.63 \times 2.914 is \dpi{120} 269.9 

Sample 2: Evaluate \dpi{120} \frac{69.2\times 426.3}{21.94\times 0.6325} .

logarithm of number example

Recall that the characteristic is \dpi{120} 3+1=4. So the result of the above evaluation is \dpi{120} 2126 .

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


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

Thanks for reading this mathematics article, hope it was educative. if you desire to write to Us or comment on this article, we do appreciate hearing from you! Click HERE to join our discussion forum. In addition, click the various links below to follow and subscribe to our various social media handles.

Join our over 20,000+ readers to receive articles on mathematics. To get information on Website design, Digital Marketing, External Examinations in Nigeria, and lots more? Visit our blog page to read articles on these and lots more.

August 13, 2021

New Track

New Track is a leading brand in educational consulting. We train Students preparing to take external examinations with academic ebooks, YouTube videos, discussion forums, etc.
We publish posts on topics ranging from the latest News for JAMB, WEAC, NECO, etc, website design, digital marketing, tropical current affairs mathematics etc. We at New Track are poised to give you the best of the best.
Look forward to seeing more students join this Community!

Leave a Reply

Your email address will not be published.