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31. The equation logax + loga(1 + x) = 0 can be written as.
  A.  x3 + x + 2
  B.  x2 + x + 1
  C.  x2 + 2x + 1
  D.  x3 + 4x + 2
     
   
View Answer

Shortcut:
log(ab) = log(a) + log(b)
Here; logax + loga(1 + x) = 0
or, logax(x + 1) = loga1 [∵ log1 = 0]
or, x(x + 1) = 1
or, x2 + x − 1 = 0

32. Find the value of log
(
a2 / bc
)
+ log
(
b2 / ac
)
+ log
(
c2 / ab
)
  A.  1
  B.  2
  C.  0
  D.  3
     
   
View Answer

Shortcut:
log(ab) = log(a) + log(b)
Here; log
(
a2 / bc
)
+ log
(
b2 / ac
)
+ log
(
c2 / ab
)

Using the shortcut, we get:
Required value = log
a2 x b2 x c2 / a2 x b2 x c2
= log1 = 0

33. If log10(m) = b + log10n, find the value of m.
  A.  b10n
  B.  2n10b
  C.  n102b
  D.  n10b
     
   
View Answer

Shortcut:
log
(
a / b
)
= log(a) − log(b)
Note: log
(
a / b
)
≠
log(m) / log(n)

Here; log10(m) = b + log10n
⇒ log10m − log10n = b
⇒ log
m / n
= b
⇒
m / n
= 10b
∴ m = n10b

34. If log102 = 0.301 then the value of log10(50) is:
  A.  1.699
  B.  2.499
  C.  1.569
  D.  1.833
     
   
View Answer

Shortcut:
log
(
a / b
)
= log(a) − log(b)
Note: log
(
a / b
)
≠
log(m) / log(n)

Here; log10(50) = log10
50 x 2 / 2

= log(100) − log2
= log(10)2 − log2
= 2 − 0.301 = 1.699

35. If log102 = 0.3010 and log107 = 0.8451, then find the value of log102.8 is:
  A.  0.4143
  B.  0.4251
  C.  0.4471
  D.  0.4391
     
   
View Answer

Shortcut:
log
(
a / b
)
= log(a) − log(b)
Note: log
(
a / b
)
≠
log(m) / log(n)

Here; log10(2.8) = log10
28 / 10

= log(28) − log(10)
= log(7 x 4) − log10
= log(7) x log(4) − log10
= log(7) x log(2)2 − log10
= log(7) + 2log2 − log10
= 0.8451 + (2 x 0.3010) − 1 = 0.4471

36. If log2 = 0.3010 then the value of log5 is:
  A.  0.6890
  B.  0.6990
  C.  0.6670
  D.  0.6950
     
   
View Answer

Shortcut:
log
(
a / b
)
= log(a) − log(b)
Note: log
(
a / b
)
≠
log(m) / log(n)

Here; log5 = log
10 / 2

= log(10) − log(2)
= 1 − 0.3010 = 0.6990

37. The simplified form of log
75 / 16
− 2log
5 / 9
+ log
32 / 343
is:
  A.  log3
  B.  log1
  C.  log4
  D.  log2
     
   
View Answer

Shortcut:
log
(
a / b
)
= log(a) − log(b)
Note: log
(
a / b
)
≠
log(m) / log(n)

Here; log
75 / 16
− 2log
5 / 9
+ log
32 / 343

= log
25 x 3 / 4 x 4
− log
25 / 81
+ log
16 x 2 / 81 x 3

= log(25 x 3) − log(4 x 4) − log(25) + log81 + log(16 x 2) − log(81 x 3)
= log25 + log3 − log16 − log25 + log81 + log16 + log2 − log81 − log3
= log2

38. If log10m = b log10n, then the value of m is:
  A.  2nb
  B.  n2b
  C.  nb
  D.  bn
     
   
View Answer

Shortcut:
logax =
logbx / logba

Here; log10m = blog10n
⇒ b = log
log10m / log10n

⇒ b = lognm
∴ m = nb

39. If log2 = 0.3010 and log3 = 0.4771, then what value of x satisfies the equation 3x+3 = 135 (approx)?
  A.  1.5
  B.  3.5
  C.  2.5
  D.  2.4
     
   
View Answer

Shortcut:
logax =
logbx / logba

Here, we have: 3x + 3 = 135
or, 3x x 33 = 135
∴ 3x =
135 / 27
= 5
⇒ log35 = x
⇒
log105 / log103
= x (by shortcut method)
⇒
log10(10/2) / log103
= x
⇒
log10 − log102 / log103
= x
⇒
1 − 0.3010 / 0.4771
= x
∴ x = 1.5 (Approx.)

40. Find the value of logba logcb logac = ?
  A.  4
  B.  1
  C.  3
  D.  2
     
   
View Answer

Shortcut:
logax =
logbx / logba

Here, logba logcb logac
or, logba x logcb x logac
=
logca / logcb
logcb x logac
[
∵ logxy =
logzy / logzx
]

= logca x logac = 1
[
∵ logca =
1 / logac
]

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