## Rajasthan Board RBSE Class 11 Maths Chapter 9 Logarithms Ex 9.3

Question 1.

Find the characteristic of logarithm of following numbers :

(i) 1270

(ii) 20.125

(iii) 7.985

(iv) 431.5

(v) 0.02

(vi) 0.02539

(vii) 70

(viii) 0.000287

(ix) 0.005

(x) 0.00003208

(xi) 0.000485

(xii) 0.007

(xiii) 0.0005309

Solution:

(i) Number 1270 is 4 digit number.

So, characteristic of its logarithm will be 4 – 1 = 3.

(ii) In 20.125, integral part is 20 which contains 2 digit.

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

(iii) In 7.985, integral part is 7 which contains 1 digit.

So, characteristic of its logarithm will be 1 – 1 = 0.

(iv) In 431.5, integral part is 431 which contains 3 digit.

So, characteristic of its logarithm will be 3 – 1 = 2.

(v) In 0.02, there are 1 zero between decimal point and first significant digit.

So, characteristic of its logarithm will be – (1 + 1) = – 2 or \(\overline { 2 }\) .

(vi) In 0.02539, there are zero between decimal point and first significant digit.

So, characteristic of its logarithm will be – (1 + 1) = – 2 or \(\overline { 2 }\).

(vii) Number 70 is 2 digit number.

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

(viii) In 0.000287, there are 3 zero between decimal point and first significant digit.

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

(ix) In 0.005, there are 2 zero between decimal point and first significant digit.

So, characteristic of its logarithm will be – (2 + 1) = – 3 or \(\overline { 3 }\).

(x) In 0.00003208, there are 4 zero between decimal point and first significant digit.

So, characteristic of its loga¬rithm will be – (4 + 1) = – 5 or \(\overline { 5 }\) .

(xi) In 0.000485, there are 3 zero between decimal point and first significant digit.

So, characteristic of its logarithm will be – (3 + 1) = -4 or \(\overline { 4 }\).

(xii) In 0.007, there are 2 zero between decimal point and first significant digit.

So, characteristic of its logarithm will be – (2 + 1) = – 3 or \(\overline { 3 }\).

(xiii) In 0.0005309, there are 3 zero between decimal point and first significant digit.

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

Question 2.

Find the logarithm of the following numbers by using log table :

(i) 2813

(ii) 400

(iii) 27.28

(iv) 9

(v) 0.678

(vii) 0.08403

(viii) 0.000287

(ix) 1.234

(x) 0.00003258

(xi) 0.000125

(xii) 0.00003208

Solution:

(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 log_{10 }2813 = 0.4492

Thus, log_{10 }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 log_{10} 400 = 0.6021

Thus, log_{10} 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 log_{10} 27.28 = 0.4359

Thus, log_{10 }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, log_{10} 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 num-ber which is 8312.

So, mantissa of log_{10} 0.678 = 0.8312

Thus,

log_{10} 0.678 = Characteristic + Mantissa

= \(\overline { 1 }\) + 0.8312 = \(\overline { 1 }\). 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 = \(\overline { 3 }\)

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 log_{10}0.0035 = 0.5441

Thus, log_{10} 0.0035 = Characteristic + Mantissa

= \(\overline { 3 }\) + 0.5441 = \(\overline { 3 }\).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 num-ber, which is = 9243

For fourth digit 3 mean difference = 2

On adding = 9245

So, mantissa of log_{10}0.08403 = 0.9245

Thus,

log_{10} 0.08403 = Characteristic + Mantissa

= \(\overline { 2 }\) + 0.9245 = \(\overline { 2 }\).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 = \(\overline { 4 }\)

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 log_{10} 0.000287 = 0.4579

Thus,

log_{10} 0.000287 = Characteristic + Mantissa

= \(\overline { 4 }\) + 0.4579

= \(\overline { 4 }\).4579

(ix) 1.234

Characteristic = 1- 1 = 0

Mantissa = 0.0899 + 14

= 0.0913

log_{10}1.234 = 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 = 5

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

For fourth digit 8 mean difference = 11

On adding = 5130

So, mantissa of log_{10} 0.00003258 = 0-5130

Thus,

log_{10} 0.00003258 = Characteristic + Mantissa

= \(\overline { 5 }\) + 0.5130

= \(\overline { 5 }\).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 = \(\overline { 4 }\)

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

So, mantissa of log_{10}0.000125 = 0.0969

Thus,

log_{10}o 0.000125 = Characteristic + Mantissa

= \(\overline { 4 }\) + 0.0969

= \(\overline { 4 }\).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 num-ber which is = 5051

For fourth digit 8’s mean difference = 11

On adding = 5062

So, mantissa of log_{10}0.0003208 = 0.5062

Thus,

log_{10}0.0003208 = Characteristic + Mantissa = \(\overline { 5 }\) + 0.5062

= \(\overline { 5 }\) .5062

#### RBSE Solutions for Class 11 Maths