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**LOGARITHMS**

**Calculation of Mantissa:**

Mantissa is the 2^{nd} part of the logarithm of a number, which is a positive proper fraction. It is calculated with the help of the log table which is usually given at the end of the book. Following are the rules to calculate mantissa as well as the method to read logarithm table:

- Unconsidered the decimal point of a given number. For example, read 23.24 as 2324. After neglecting the decimal point, if the number of digits is not four, then make them four by inserting ciphers to the right side of a given number accordingly. Such as, read 10.6 as 1060 or, read 2.5 as 2500.

[*If the number of digits after neglecting the decimal point is more than four, say five digits, then add 1 to the 4*]^{th}figure if the 5^{th}figure is equal to or greater than 5 or leave it, in case the 5^{th}figure is less than 5. - Take the first 2 digits of the number i.e. 23 from 2324 and run the search down the extreme left-hand column of the log table and stop at the number consisting of those first 2 digits.
- Take the third digit of the number i.e. 2 and read the intersection of horizontal row beginning with 23 and vertical column headed by 2, which gives the number 3655.
- Now, take the fourth digit which is 4 and read the intersection of horizontal row beginning with 23 and vertical column headed by 4 in the mean difference column, which gives the number 7.
- Then, by adding the number obtained in
to the number obtained in__d__, we get 3655 + 7 = 3662.__c__ - Now, if we prefix the decimal to the number obtained by
, the mantissa we get of the number 23.24 is 0.3662.__e__

After that, it is observed that the mantissa of the logarithm of a number remains unchanged if the position of the decimal point is changed without any change in the digits and order of digits.

**Calculation of Logarithm of a Number:**

The logarithm of a number is calculated by finding out its characteristic and mantissa and adding them up.

Logarithm = Characteristic + Mantissa

**Examples:**

Use log table and find the followings’ values:

- log 189.5
- log 2214
- log 6
- log 0.04205

**Solution:**

__log 189.5__

Here, 189.5 has 3 digits as its integral part i.e. 189.

Now, according to the rule,

the characteristic = 3 – 1 = 2

where the mantissa is 0.2777.

Hence,

log 189.5 = 2 + 0.2777 = 2.2777__log 2214__Here, 2214 has 4 digits as its integral part i.e. 2214.

Now, according to the rule,

the characteristic = 4 – 1 = 3

where the mantissa is 0.3452.

Hence,

log 2214 = 3 + 0.3452 = 3.3452__log 6__Here, the number 6 (6.000) has 1 digit in its integral part i.e. 6.

Now, according to the rule,

the characteristic = 1 – 1 = 0

where the mantissa is 0.7782.

Hence,

log 6 = 0 + 0.7782 = 0.7782__log 0.04205__Here, the number 0.04205 is less than unity and immediately, there is only 1 cipher after the decimal point and before the significant figure 4.

Now, according to the rule,

the characteristic = – 1 – 1 = –2

where for mantissa , shift the decimal point and read the number as 0.4205 now.

Therefore, the mantissa is 0.6237.

Hence,log 0.04205 = –2 + 0.6237 = \(\overline {2}\).6237*Note: \(\overline {2}\).6237 does not mean –2.6237. \(\overline {2}\).6237 means that –2 is the characteristic and +0.6237 is the mantissa.*

**Antilogarithm:**

If log_{e} N = x, it means that **N** is the antilogarithm of **x**, to the base **e**. In short, it is written as antilog_{e} x. As logarithm table, the table of antilogarithm is also given at the end of the book.

**Examples:**

Find the numbers whose antilogarithms are:

- 4201
- \(\overline {2}\).6237

**Solution:**

- Given,

log N = 1.4201

Consider the decimal part 0.4201 and read the antilogarithm table in the same manner as logarithm table. Take the first 2 digits of the number i.e. 42 from 4201 and run the search down the extreme left-hand column of the log table and stop at the number 42.

Now, take the third digit of the number i.e. 0 and read the intersection of horizontal row beginning with .42 and vertical column headed by 0 from the left, which gives the number 2630.

Again, take the fourth digit which is 1 and read the intersection of horizontal row beginning with .42 and vertical column headed by 1 in the mean difference column, which gives the number 1.

Then, by adding the numbers obtained above, we get 2630 + 1 = 2631.

Now, consider the integral part i.e. 1 which is greater than zero. Thus, add 1 to it, after which we get 2.

Therefore, finally, prefix the decimal point after 2 digits in the number obtained above which is 2631.

Hence, the required number whose antilog is 1.4201 is 26.31. - Given,
log N = \(\overline {2}\).6237

Consider the decimal part 0.6237 and take the first 2 digits of the number i.e. 62 from 6237 and run the search down the extreme left-hand column of the log table and stop at the number 62.

Now, take the third digit of the number i.e. 3 and read the intersection of horizontal row beginning with .62 and vertical column headed by 3 from the left, which gives the number 4198.Again, take the fourth digit which is 7 and read the intersection of horizontal row beginning with 62 and vertical column headed by 7 in the mean difference column, which gives the number 7.

Then, by adding the numbers obtained above, we get 4198 + 7 = 4205.

Now, consider the integral part which is less than zero and negative. Thus, subtract 1 from the number under the bar sign viz. 2 – 1 = 1 and attach one cipher to the left-hand side of the number 4205. Finally, prefix the decimal point.

Hence, the required number whose antilog is \(\overline {2}\).6237 is 0.04205.

(Dhakal & Ghimire, 2016)

Bibliography

Dhakal, B., & Ghimire, J. L. (2016). *Business Mathematics.* Putalisadak: Asmita Publication.

Logarithms:

- Calculation of Mantissa

Antilogarithm

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