How to use log table – Examples

How to use log table 

A logarithm is the inverse of the exponential function.

a^{b}=\text{x}
⇒ \log _{a} {\text x}=b

Here, the logarithm of x with base a is equal to b.

It is mainly classified into 3 types based on the value of the base.

Common logarithm of \text{x}=\log _{10} \text{x}(\text { Base }=10)

Natural logarithm of \text{x}=\log _{e} \text{x}(\text { Base }=e)

Binary logarithm of \text{x}=\log _{2} \text{x}(\text { Base }=2)

Logarithm of x = Characteristic of x + Mantissa of x

Characteristic is an integer whereas mantissa is a decimal number.

Finding Characteristic of x

1. Write the number in scientific form.

For example, \left(15.53=1.553 \times 10^{1}\right) \text { and }\left(0.0045=4.5 \times 10^{-3}\right)

2. The power of base is the characteristic of x.

Finding the Mantissa of x

1. Different log tables are used for common logarithm and Natural logarithm depending on the base of the logarithm.

2. Search for the row number corresponding to the first two digits of x.

Search for the column number corresponding to the third digit of x.

Note the value at the intersection of the row and column given above.

3. Search for the row number corresponding to the first two digits of x.

Search for the row number corresponding to the fourth digit of x on the mean difference table.

Note the value at the intersection of the row and column given above.

4. Add the values obtained in step 2 and step 3 to obtain the mantissa.

Finding the value of the logarithm

Logarithm of x = Characteristic of x + Mantissa of x

Solved Examples

1. Find the value of \log _{10} 15.53.

Solution:

Finding Characteristic 

Write the number in scientific form.

The power of base is the characteristic. 

Here the base is 10 and the power of 10 is 1. Hence,1 is the characteristic.

Finding the Mantissa 

1. Different log tables are used for common logarithm and Natural logarithm depending on the base of the logarithm. We have to refer to the common logarithm table as the base is 10.

2. Search for the row number corresponding to the first two digits 15.

Search for the column number corresponding to the third digit of 5.

Note the value at the intersection of row 15 and column 5.

The value is 1903.

3. Search for the row number corresponding to the first two digits 15.

Search for the row number corresponding to the fourth digit 3 on the mean difference table.

Note the value at the intersection of row 15 and column 3 of the mean difference table. 

The value is 8.

4. Add the values obtained in step 2 and step 3 to obtain the mantissa.

1903 + 8 = 1911

Hence the mantissa of 15.53 is 0.1911.

Value of the logarithm

Logarithm of x = Characteristic of x + Mantissa of x

\log _{10} 15.53=1+0.1911=1.1911

2. Find the value of \log _{10} 0.243.

Solution:

Finding Characteristic 

Write the number in scientific form.

0.243=2.43 \times 10^{-1}

The power of base is the characteristic 

Here the base is 10 and the power of 10 is -1. Hence, (-1) is the characteristic.

Finding the Mantissa 

1. Different log tables are used for common logarithm and Natural logarithm depending on the base of the logarithm. We have to refer to the common logarithm table as the base is 10.

2. Search for the row number corresponding to the first two digits 24.

Search for the column number corresponding to the third digit of 3.

Note the value at the intersection of row 24 and column 3.

The value is 3856.

There is no fourth digit. So we have to stop here.

Hence the mantissa of 0.243 is 0.3856.

Value of the logarithm

Logarithm of x = Characteristic of x + Mantissa of x

\log _{10} 0.243=(-1)+0.3856=(-0.6144)