(i) 2813

Characteristic : There are characteristic will be 4 – 1 = 3.

Mantissa : In log table, in first column, in front of 28 and under the column 1.

Find the number which is = 4487.

For fourth digit 3’s mean difference = 5

On adding = 4492

So, mantissa of log₁₀ 2813 = 0.4492

Thus, log₁₀ 2813 = Characteristic + Mantissa

= 3 + 0.4492 = 3.4492

(ii) 400

Characteristic : There are 3 digit in number 400.

So, characteristic will be 3 – 1 = 2.

Mantissa : In log table, in first column, in front of 40 and under the column 0, find the number which is 6021.

So, mantissa of log₁₀ 400 = 0.6021

Thus, log₁₀ 400 = Characteristic + Mantissa

= 2 + 0.6021 =2.6021

(iii) 27.28

Characteristic : 

Here, integral part is 27 which contains 2 digit. 

So, characteristic of its logarithm will be 2 – 1 = 1.

Mantissa : In log table, in first column, in front of 27 (ignoring decimal point) and under the column 2, find the number which is = 4346

For fourth digit 8’s mean difference = 13

On adding = 4359

So, mantissa of log₁₀ 27.28 = 0.4359

Thus, log₁₀ 27.28 = Characteristic + Mantissa

= 1+ 0.04359 =1.4359

(iv) 9

Characteristic : There are only 1 digit in number 9.

So, characteristic will be 1 – 1 = 0.

Mantissa : In log table, in first column, in front of 90 and under the column 0 find the number which is = 9542.

So mantissa of log₁₀ 9 = 0.9542

Thus, log₁₀ 9 = Characteristic + Mantissa

= 0 + 0.9542 = 0.9542

(v) 0-678

Characteristic : In 0.678, there are no zero between decimal point and first significant digit. 

So, characteristic of its logarithm will be – (0+ 1) = – 1 = 1

Mantissa : In log table, in first column, in front of 67 (ignoring decimal point) and under the column 8 find the number which is 8312.

So, mantissa of log₁₀ 0.678 = 0.8312
Thus,
log₁₀ 0.678 = Characteristic + Mantissa

= \(\bar{1}\) + 0.8312 = \(\bar{1}\). 8312

(vi) 0.0035

Characteristic : In 0.0035, there are 2 zero between decimal point and first significant digit. 

So, characteristic of its logarithm will be
-(2+1) = -3 = \(\bar{3}\)

Mantissa : In log table, in first column, in front of 35 (ignoring decimal point) and under the column 0, find the number which is = 5441

So, mantissa of log₁₀0.0035 = 0.5441

Thus, log₁₀ 0.0035 = Characteristic + Mantissa

= \(\bar{3}\) + 0.5441 = \(\bar{3}\).5441

(vii) 0.08403

Characteristic : In 0.08403, there are 1 zero between decimal point and first significant digit. 

So, characteristic of its logarithm will be
-(1 + 1) = – 2 = 2

Mantissa : In log table, in first column, in front of 84 (ignoring decimal point) and under the column 0 find the number, which is = 9243

For fourth digit 3 mean difference = 2

On adding = 9245

So, mantissa of log₁₀0.08403 = 0.9245

Thus,
log₁₀ 0.08403 = Characteristic + Mantissa

= \(\bar{2}\) + 0.9245 = \(\bar{2}\).9245.

(viii) 0.000287

Characteristic: In 0.000287, there are 3 zero between decimal point and first significant digit. 

So, characteristic of its logarithm will be
-(3 + 1) = -4 = \(\bar{4}\)

Mantissa : In log table, in first column, in front of 28 (ignoring decimal point) and under the column 7 find the number which is 4579.

So, mantissa of log₁₀ 0.000287 = 0.4579

Thus,

log₁₀ 0.000287 = Characteristic + Mantissa

= \(\bar{4}\) + 0.4579

= \(\bar{4}\).4579

(ix) 1.234

Characteristic = 1- 1 = 0

Mantissa = 0.0899 + 14

= 0.0913

log₁₀1.234 = Characteristic + Mantissa

= 0 + 0.0913

= 0.0913

(x) 0.00003258

Characteristic: In 0.00003258, there are 4 zero between decimal point and first significant digit. 

So characteristic of its logarithm will be
-(4 + 1) = -5 = \(\bar{5}\)

Mantissa : In log table, in first column, in front of 32 (ignoring decimal point) and under the column 5 find the number, which is = 5119

For fourth digit 8 mean difference = 11

On adding = 5130

So, mantissa of log₁₀ 0.00003258 = 0-5130
Thus,

log₁₀ 0.00003258 = Characteristic + Mantissa

= \(\bar{5}\) + 0.5130

= \(\bar{5}\).5130

(xi) 0.000125

Characteristic : In 0.000125, there are 3 zero between decimal point and first significant digit. 

So, characteristic of its logarithm will be
– (3 + 1) = – 4 = \(\bar{4}\)

Mantissa : In log table, in first column, in front of 12 (ignoring decimal point) and under the column 5 find the number which is = 0969

So, mantissa of log₁₀0.000125 = 0.0969
Thus,

log₁₀o 0.000125 = Characteristic + Mantissa

= \(\bar{4}\) + 0.0969

= \(\bar{4}\).0969

(xii) 0.00003208

Characteristic: In 0-00003208, there are 4 zero between decimal point and first significant digit. 

So, characteristic of its logarithm will be
-(4 + 1) = -5 = 5

Mantissa : In log table, in first column, in front of 32 (ignoring decimal point) and under the column 0 find the number which is = 5051

For fourth digit 8’s mean difference = 11

On adding = 5062

So, mantissa of log₁₀0.0003208 = 0.5062
Thus,

log₁₀0.0003208 = Characteristic + Mantissa = \(\bar{5}\) + 0.5062

= \(\bar{5}\) .5062