Exercise 5
Q.1. Write the characteristic of the following logarithms.
i. log 6987
Solution: 4-1=3 (Hint: For whole number = n-1, where n is the total number of digits)
ii. log 256987
Solution: 6-1=5 (Hint: For whole number = n-1, where n is the total number of digits)
iii. log 0.000089
Solution: -(4+1)=-5 (Hint: For Only decimal number = -(n+1), where n is the total number of consecutive zeroes after decimal from right to left)
iv. log 4.68
Solution: 1-1=0 (Hint: For whole number = n-1, where n is the total number of digits on whole number side)
v. log 0.7491
Solution: -(0+1)=-1 (Hint: For only decimal number = -(n-1), where n is the total number of consecutive zeroes after decimal)
vi. log 1
Solution: 1-1=0 (Hint: For whole number = n-1, where n is the total number of digits)
vii. log 35.492
Solution: 2-1=1 (Hint: For whole number = n-1, where n is the total number of digits)
viii. log 36
Solution: 2-1=1 (Hint: For whole number = n-1, where n is the total number of digits)
ix. log 0.00305
Solution: -(2+1)=-3 (Hint: For only decimal number = -(n-1), where n is the total number of consecutive zeroes after decimal)
x. log 1.965
Solution: 1-1=0 (Hint: For whole number = n-1, where n is the total number of digits)
Q.2. If log 2925=3.46612 then find the values of log 29.25, log 2.925, log 0.002925 and log 292500.
Solution: Given log 2925=3.46612
∴ log 29.25 = log (2925 x 10-2)
(Hint: log axb=log a + log b)
= log 2925 + log 10-2
(Hint: log ax = xlog a)
= 3.46612 + (-2)log 10
(Hint: log 10 = 1)
= 3.46612 -2 x 1
=3.46612-2
= 1.4.6612
∴ log 2.925=log (2925 x 10-3)
(Hint: log axb=log a + log b)
= log 2925 + log 10-3
(Hint: log ax = xlog a)
= 3.46612 + (-3)log 10
(Hint: log ax = xlog a)
= 3.46612 -3 x 1
= 3.46612 - 3
= 0.46612
∴ log 0.002925=log (2925 x 10-6)
(Hint: log axb=log a + log b)
= log 2925 + log 10-6
(Hint: log ax = xlog a)
= 3.46612 + (-6)log 10
(Hint: log ax = xlog a)
= 3.46612 -6 x 1
= 3.46612 - 6
= -3.46612
∴ log 292500=log (2925 x 102)
(Hint: log axb=log a + log b)
= log 2925 + log 102
(Hint: log ax = xlog a)
= 3.46612 + 2log 10
(Hint: log ax = xlog a)
= 3.46612 + 2 x 1
= 3.46612 + 2
= 5.46612
Q.3. Find antilogarithm with the help of table.
i. 3.1465
Solution: Antilog (3.1465)= Antilog (3 + 0.1465)
=Value of 0.1465 in antilog table will be 14012
= Here characteristic of given number is positive i.e. n=3 so we have to put decimal after n+1 from left side. Here n+1= 3+1=4. Hence decimal will be after 4 digits from left side.
∴ Antilog (3.1465)= 1401.2
ii. 1 .8621
Solution: Antilog (1.8621)= Antilog (-1+0.8621)
=Value of 0.8621 in antilog table will be 72795
= Here characteristic of given number is negative i.e. n=-1 so we have to put n+1 zeroes after decimal and before number. Here n+1= -1+1=0. Hence we have to put zero zeroes after decimal and before number.
∴ Antilog (1.8621)= 0.72795
iii. 4.6663
Solution: Antilog (4.6663)= Antilog (-4+0.6663)
=Value of 0.6663 in antilog table will be 46377
= Here characteristic of given number is negative i.e. n=-4 so we have to put n+1 zeroes after decimal and before number. Here n+1= -4+1=-3. Hence we have to put 3 zeroes after decimal and before number.
∴ Antilog (4.6663)= 0.00046377
iv. -2.7917
Solution: Antilog (-2.7917)= Antilog (-2-0.7917)
(Adding 1 and subtracting 1)
= Antilog (-2+1-1-0.7917)
(Rearranging)
= Antilog {(-2-1)+(1-0.7917)}
= Antilog (-3+0.2083)
=Value of 0.2083 in antilog table will be 16155
= Here characteristic of given number is negative i.e. n=-3 so we have to put n+1 zeroes after decimal and before number. Here n+1= -3+1=-2. Hence we have to put 2 zeroes after decimal and before number.
∴ Antilog (-2.7917)= 0.0016155
v. 2.5591
Solution: Antilog (2.5591)= Antilog (2 + 0.5591)
=Value of 0.5591 in antilog table will be 36232
= Here characteristic of given number is positive i.e. n=2 so we have to put decimal after n+1 from left side. Here n+1= 2+1=3. Hence decimal will be after 3 digits from left side.
∴ Antilog (2.5591)= 362.32
Q.4. Given that log 2=0.30103, log 3=0.47712 and log 5=0.69897. Find the values of
i. log 0.002
Solution: log 0.002 = log (2 x 10-3)
(Hint: log axb=log a + log b)
= log 2 + log 10-3
(Hint: log ax = xlog a)
= 0.30103 + (-3)log 10
(Hint: log ax = xlog a)
= 0.30103 - 3 x 1
= 0.30103 -3
= -3 + 0.30103
= -3.30103
ii. log 25
Solution: log 25 = log (5 x 5)
(Hint: log axb=log a + log b)
= log 5 + log 5
(Hint: log 5=0.69897)
= 0.69897 + 0.69897
= 1.39794
iii. log 7.2
Solution: log 7.2 = log (72 x 10-1)
(Hint: log axb=log a + log b)
= log 72 + log 10-1
(Hint: log 10x = x )
= log (8 x 9) + (-1)
(Hint: log axb=log a + log b)
= log 8 + log 9 -1
= log 23 + log 32 -1
(Hint: log ax = xlog a )
= 3log 2 + 2log 3 -1
(Hint: log 2=0.30103, log 3=0.47712)
= 3 x 0.30103 + 2 x 0.47712 -1
= 0.90309 + 0.95424 -1
= 1.85733 - 1
= 0.85733
iv. log 300
Solution: log 300 = log (3 x 102)
(Hint: log axb=log a + log b)
= log 3 + log 102
(Hint: log 3=0.47712, log 10x = x )
= 0.47712 + 2
= 2.47712
iv. log 0.15
Solution: log 0.15 = log (15 x 10-2)
(Hint: log axb=log a + log b)
= log 15 + log 10-2
= log (3 x 5) - 2
= log 3 + log 5 - 2
( Hint: log 3=0.47712, log 5=0.69897)
= 0.47712 + 0.69897 - 2
(Hint: Add and subtract 1)
= 1.17609 -2 + 1 - 1
(Hint: Rearranging)
= (-2 + 1) + 1.17609 - 1
= -1 + 0.17609
= -1.17609
= 1.17609
Q.5. Find the number of digits of the following numbers.
i. 220
Solution: Let x = 220
Taking log on both side
⇒ log x = log 220
⇒ log x = 20log 2
(Hint: log 2=0.30103 )
⇒ log x = 20 x 0.30103
⇒ log x = 6.0206
(Characteristic + Mantissa)
⇒ log x = 6 + 0.0206
ii. 225
Solution: Let x = 225
Taking log on both side
⇒ log x = log 225
⇒ log x = 25log 2
(Hint: log 2=0.30103 )
⇒ log x = 25 x 0.30103
⇒ log x = 7.52575
(Characteristic + Mantissa)
⇒ log x = 7 + 0.52575
iii. 317
Solution: Let x = 317
Taking log on both side
⇒ log x = log 317
⇒ log x = 17log 3
( Hint: log 3=0.47712)
⇒ log x = 17 x 0.47712
⇒ log x = 8.11104
(Characteristic + Mantissa)
⇒ log x = 8 + 0.11104
iv. 515
Solution: Let x = 515
Taking log on both side
⇒ log x = log 515
⇒ log x = 15log 5
( Hint: log 5=0.69897)
⇒ log x = 15 x 0.69897
⇒ log x = 10.48455
(Characteristic + Mantissa)
⇒ log x = 10 + 0.48455
v. 620
Solution: Let x = 620
Taking log on both side
⇒ log x = log 620
⇒ log x = 20log 6
⇒ log x = 20log (2 x 3)
⇒ log x = 20(log 2 + log 3)
(Hint: log 2=0.30103, log 3=0.47712)
⇒ log x = 20(0.30103 + 0.47712)
⇒ log x = 20 x 0.77815
⇒ log x = 15.563
(Characteristic + Mantissa)
⇒ log x = 15 + 0.563
vi. 713
Solution: Let x = 713
Taking log on both side
⇒ log x = log 713
⇒ log x = 13log 7
(Hint: log 7=0.8451)
⇒ log x = 13 x 0.8451
⇒ log x = 10.9863
(Characteristic + Mantissa)
⇒ log x = 10 + 0.9863
vii. 2200 x 310
Solution: Let x = 2200 x 310
Taking log on both side
⇒ log x = log (2200 x 310)
⇒ log x = log 2200 + log 310
⇒ log x = 200log 2 + 10log 3
(Hint: log 2=0.30103, log 3=0.47712)
⇒ log x = 60.206 + 4.7712
⇒ log x = 64.9772
(Characteristic + Mantissa)
⇒ log x = 64 + 0.9772
viii. 312 x 28
Solution: Let x = 312 x 28
Taking log on both side
⇒ log x = log (312 x 28)
⇒ log x = log 312 + log 28
⇒ log x = 12log 3 + 8log 2
(Hint: log 2=0.30103, log 3=0.47712)
⇒ log x = 5.72544 + 2.40824
⇒ log x = 8.13368
(Characteristic + Mantissa)
⇒ log x = 8 + 0.13368
ix. 2100
Solution: Let x = 2100
Taking log on both side
⇒ log x = log 2100
⇒ log x = 100log 2
(Hint: log 2=0.30103)
⇒ log x = 30.103
(Characteristic + Mantissa)
⇒ log x = 30 + 0.103
x. 618
Solution: Let x = 618
Taking log on both side
⇒ log x = log 618
⇒ log x = 18log 6
⇒ log x = 18log (2 x 3)
⇒ log x = 18(log 2 + log 3)
(Hint: log 2=0.30103, log 3=0.47712)
⇒ log x = 18(0.30103 + 0.47712)
⇒ log x = 18 x 0.77815
⇒ log x = 14.0067
(Characteristic + Mantissa)
⇒ log x = 14 + 0.0067
= Here characteristics of x is 14 i.e. n=14
Q.6. How many zeroes are there between the decimal point and the first significant digit after decimal in the following numbers.
i. 2-64
Solution: Let x = 2-64
Taking log on both side
⇒ log x = log 2-64
⇒ log x = -64log 2
⇒ log x = -64 x 0.30103
⇒ log x = -19.26592
⇒ log x = -19 - 0.26592
(Add +1 -1)
⇒ log x = -19 - 1 + 1 - 0.26592
⇒ log x = (-19 - 1) + (1 - 0.26592)
(Characteristic + Mantissa)
⇒ log x = -20 + 0.73408
= Here characteristics of x is 20 i.e. n= 20
Taking log on both side
⇒ log x = 1000(log 1 - log 2)
(Hint: log 1=0, log 2=0.30103)
⇒ log x = 1000 x (0 - 0.30103)
⇒ log x = 1000 (-0.30103)
⇒ log x = -301.03
⇒ log x = -301 - 0.03
(Add +1 -1)
⇒ log x = -301 - 1 + 1 - 0.03
⇒ log x = (-301 - 1) + (1 - 0.03)
(Characteristic + Mantissa)
⇒ log x = -302 + 0.97
= Here characteristics of x is 302 i.e. n= 302
iii. 3-6
Solution: Let x = 3-6
Taking log on both side
⇒ log x = log 3-6
⇒ log x = -6log 3
(Hint: log 3=0.47712)
⇒ log x = -6 x 0.47712
⇒ log x = -2.86272
⇒ log x = -2 - 0.86272
(Add +1 -1)
⇒ log x = -2 - 1 + 1 - 0.86272
⇒ log x = (-2 - 1) + (1 - 0.86272)
(Characteristic + Mantissa)
⇒ log x = -3 + 0.13728
= Here characteristics of x is 3 i.e. n= 3
Taking log on both side
⇒ log x = 100(log 1 - log 3)
(Hint: log 1=0, log 3=0.47712)
⇒ log x = 100 x (0 - 0.47712)
⇒ log x = 100 (-0.47712)
⇒ log x = -47.712
⇒ log x = -47 - 0.712
(Add +1 -1)
⇒ log x = -47 - 1 + 1 - 0.712
⇒ log x = (-47 - 1) + (1 - 0.712)
(Characteristic + Mantissa)
⇒ log x = -48 + 0.288
= Here characteristics of x is 48 i.e. n= 48
v. (0.0016)20
Solution: Let x = (0.0016)20
Taking log on both side
⇒ log x = log (0.0016)20
⇒ log x = 20log (0.0016)
⇒ log x = 20log (16 x 10-4)
⇒ log x = 20(log 24 + log10-4)
⇒ log x = 20(4log 2 -4log10)
(Hint: log 2=0.30103)
⇒ log x = 20(4 x 0.30103 -4)
⇒ log x = 20(1.20412 -4)
⇒ log x = 20(-2.79588)
⇒ log x = -55.9176
⇒ log x = -55 - 0.9176
(Add +1 -1)
⇒ log x = -55 - 1 + 1 - 0.9176
⇒ log x = (-55 - 1) + (1 - 0.9176)
(Characteristic + Mantissa)
⇒ log x = -56 + 0.0824
= Here characteristics of x is 56 i.e. n= 56
vi. 3-7
Solution: Let x = 3-7
Taking log on both side
⇒ log x = log 3-7
⇒ log x = -7log 3
(Hint: log 3=0.47712)
⇒ log x = -7 x 0.47712
⇒ log x = -3.33984
⇒ log x = -3 - 0.33984
(Add +1 -1)
⇒ log x = -3 - 1 + 1 - 0.33984
⇒ log x = (-3 - 1) + (1 - 0.33984)
(Characteristic + Mantissa)
⇒ log x = -4 + 0.66106
= Here characteristics of x is 4 i.e. n= 4
vii. 16.8-12
Solution: Let x = 16.8-12
Taking log on both side
⇒ log x = log 16.8-12
⇒ log x = -12log 16.8
⇒ log x = -12log (168 x 10-1)
⇒ log x = -12{log (23x 3 x 7) -1 log10}
⇒ log x = -12{log (23x 3 x 7) -1}
⇒ log x = -12{3log2 + log3 + log7 -1}
⇒ log x = -12{3log2 + log3 + log7 -1}
⇒ log x = -12{3 x 0.30103 + 0.47712 + 0.84510 -1}
⇒ log x = -12{2.22531 - 1}
⇒ log x = -12 x 1.22531
⇒ log x = -14.70372
⇒ log x = -14 - 0.70372
(Add +1 -1)
⇒ log x = -14 - 1 + 1 - 0.70372
⇒ log x = (-14 - 1) + (1 - 0.70372)
(Characteristic + Mantissa)
⇒ log x = -15 + 0.70372
= Here characteristics of x is 15 i.e. n= 15
viii. 7-4
Solution: Let x = 7-4
Taking log on both side
⇒ log x = log 7-4
⇒ log x = -4log 7
(Hint: log7= 0.84510)
⇒ log x = -4 x 0.84510
⇒ log x = -3.3804
⇒ log x = -3 - 0.3804
(Add +1 -1)
⇒ log x = -3 - 1 + 1 - 0.3804
⇒ log x = (-3 - 1) + (1 - 0.3804)
(Characteristic + Mantissa)
⇒ log x = -4 + 0.3804
= Here characteristics of x is 4 i.e. n= 4
ix. 3-21
Solution: Let x = 3-21
Taking log on both side
⇒ log x = log 3-21
⇒ log x = -21log 3
(Hint: log3= 0.47712)
⇒ log x = -21 x 0.47712
⇒ log x = -10.01952
⇒ log x = -10 - 0.01952
(Add +1 -1)
⇒ log x = -10 - 1 + 1 - 0.01952
⇒ log x = (-10 - 1) + (1 - 0.01952)
(Characteristic + Mantissa)
⇒ log x = -11 + 0.01952
= Here characteristics of x is 11 i.e. n= 11
x. 2-10
Solution: Let x = 2-10
Taking log on both side
⇒ log x = log 2-10
⇒ log x = -10log 2
(Hint: log2= 0.30103)
⇒ log x = -10 x 0.30103
⇒ log x = -3.0103
⇒ log x = -3 - 0.0103
(Add +1 -1)
⇒ log x = -3 - 1 + 1 - 0.0103
⇒ log x = (-3 - 1) + (1 - 0.0103)
(Characteristic + Mantissa)
⇒ log x = -4 + 0.0103
= Here characteristics of x is 4 i.e. n= 4
∴ So the number of zeroes between the decimal point and the first significant digit after decimal= n - 1 = 4 - 1 = 3
Q. Find the values:
i. (7.92)-5
Solution: Let x = (7.92)5
Taking log on both side
⇒ log x = log (7.92)5
⇒ log x = 5log 7.92
(Hint: From Log table log 7.92= 0 + 0.89873)
⇒ log x = 5 (0 + 0.89873)
⇒ log x = 4.49365
⇒ x = Antilog (4.49365)
(Hint: From Antilog table antilog (4.4936) = 31162.05)
∴ x = 31162.05
i. ∛0.00000165
Solution: Let x = ∛0.00000165
Taking log on both side
⇒ log x = log ∛0.00000165
⇒ log x = log (0.00000165)
(Hint: From Log table log 7.92= 0 + 0.89873)
⇒ log x = 5 (0 + 0.89873)
⇒ log x = 4.49365
⇒ x = Antilog (4.49365)
(Hint: From Antilog table antilog (4.4936) = 31162.05)
∴ x = 31162.05