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 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 num-ber which is 8312.
So, mantissa of log10 0.678 = 0.8312
Thus,
log10 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 log100.0035 = 0.5441
Thus, log10 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 log100.08403 = 0.9245
Thus,
log10 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 log10 0.000287 = 0.4579
Thus,
log10 0.000287 = Characteristic + Mantissa
= \(\overline { 4 }\) + 0.4579
= \(\overline { 4 }\).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 = 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 log10 0.00003258 = 0-5130
Thus,
log10 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 log100.000125 = 0.0969
Thus,
log10o 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 log100.0003208 = 0.5062
Thus,
log100.0003208 = Characteristic + Mantissa = \(\overline { 5 }\) + 0.5062
= \(\overline { 5 }\) .5062
RBSE Solutions for Class 11 Maths