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41. Given that log102 = 0.3010, then log210 is equal to:
  A.  
1000 / 301
  B.  
1052 / 299
  C.  
1121 / 304
  D.  
1050 / 297
     
   
View Answer

Shortcut:
logax =
logbx / logba

Here, log210 =
log10 / log2

=
1 / log2

=
1.0000 / 0.3010
=
1000 / 301

42. If log12 = a then log616 is:
  A.  
4(2 − a) / 4 + a
  B.  
4(3 − a) / 3 + a
  C.  
3(5 − a) / 2 + a
  D.  
6(2 − a) / 5 + a
     
   
View Answer

Shortcut:
logax =
logbx / logba

Here, log1227 = a
or,
log27 / log12
= a
or, alog12 = log33
or, alog(3 x 4) = 3log3
or, a[log3 + log4] = 3log3
or, alog4 + alog3 = 3log3
or, alog22 = (3 − a)log3
or, 2alog2 = (3 − a)log3
or,
log2 / log3
=
3 − a / 2a
-----------(Eq i)
Now, log616 =
log16 / log6
=
log24 / log(2 x 3)
=
4log2 / log2 + log3

=
4 x (log2/log3) / (log2/log3) + 1

=
4 x (3 − a/2a) / (3 − a/2a) + 1

=
4(3 − a) / 3 + a

43. If loga4 = 0.4, then the value of a is.
  A.  30
  B.  36
  C.  32
  D.  35
     
   
View Answer

Shortcut:
logax =
logbx / logba

Here, loga4 = 0.4
or,
log4 / loga
= a
or,
2log2 / loga
=
2 / 5

or, loga = 5log2 = log25 = log32
∴ a = 32

44. If logay = 100 and log2a = 10, then the value of y is:
  A.  2100
  B.  2a1000
  C.  21000
  D.  21000a
     
   
View Answer

Shortcut:
logax =
logbx / logba

Here, logay = 100 and log2a = 10
or,
log(y) / log(x)
= 100 and
logx / log2
= 10
or,
logy / log2
= 100 x 10 = 1000
or, log2y = 1000
or, y = 21000

45. If ax = b, by = c, cz = a, then find the value of xyz is:
  A.  1
  B.  3
  C.  2
  D.  4
     
   
View Answer

Shortcut:
logax =
logbx / logba

Here, ax = b, by = c, cz = a
ax = b or, logab = x
by = c or, logbc = y
cz = a or, logca = z
Now, x x y x z = logab x logbc x logca = 1

46. The value of log23 × log32 × log43 is:
  A.  2
  B.  5
  C.  3
  D.  1
     
   
View Answer

Shortcut:
logax =
logbx / logba

Here, log23 × log32 × log43
log3 / log2
x
log2 / log3
x
log4 / log3
x
log3 / log4
= 1

47. The value
log ax / log abx
− logab is:
  A.  1
  B.  3
  C.  2
  D.  4
     
   
View Answer

Shortcut:
logax =
logbx / logba

Here, logax =
logabx / logaba

∴ the given expression =
1 / logaba
− logab
= logaab − logab
= loga
ab / b

= logaa = 1

48. If log(a + 2) = log (a) + log(2), then find the value of a is:
  A.  2
  B.  1
  C.  5
  D.  4
     
   
View Answer

Shortcut:
log(a + b) = log(a) + log(b), then a =
b / b − 1

Here, a = a, b = 2
Using these values in the shortcut, we get:
a =
2 / 2 − 1
= 2
Detailed method:
log(a + 2) = log (a) + log(2) = log(2a)
or, a + 2 = 2a
∴ a = 2

49. If log(a − 2) = log a − log(2), then find the value of a is:
  A.  7
  B.  4
  C.  2
  D.  3
     
   
View Answer

Shortcut:
log(a − b) = log(a) − log(b), then a =
b2 / b − 1

Here, a = a, b = 2
Using these values in the shortcut, we get:
a =
(2)2 / 2 − 1
= 4
Detailed method:
log(a − 2) = log (a) + log(2)
or, a − 2 =
a / 2

or, 2a − 4 = a
∴ a = 4

50. Find the number of digits in 810. (Given that log102 = 0.3010)
  A.  15
  B.  11
  C.  8
  D.  10
     
   
View Answer

Shortcut:
To find the number of digits in mn.
Number of digits = [Integral part of (nlog10m)] + 1
We have: 810 = (23)10 = 230
Here, m = 2, n = 30
Using these values in the shortcut, we get:
No. of digits = [30log102) + 1]
= (30 x 0.3010) + 1
= (9.03) + 1 = 9 + 1 = 10

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