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11. Find the value of log25125 − log84
  A.  
3 / 7
  B.  
2 / 3
  C.  
5 / 6
  D.  
5 / 9
     
   
View Answer

Shortcut:
logbyax =
x / y
(logba)
If b = a = n, then
lognynx =
x / y

Here, log25125 − log84
log52(53) − log23(22)
Using the shortcut, we get:
=
3 / 2
−
2 / 3
=
5 / 6

12. Find the value of log981 − log432.
  A.  −
1 / 5
  B.  −
2 / 5
  C.  −
3 / 4
  D.  −
1 / 2
     
   
View Answer

Shortcut:
logbyax =
x / y
(logba)
If b = a = n, then
lognynx =
x / y

Here, log981 − log432
log32(34) − log22(25)
Using the shortcut, we get:
=
4 / 2
−
5 / 2
= −
1 / 2

13. Find the value of log3228 + log24337 − log361296
  A.  1
  B.  3
  C.  2
  D.  4
     
   
View Answer

Shortcut:
logbyax =
x / y
(logba)
If b = a = n, then
lognynx =
x / y


Here, log2528 + log3537 − log6264
Using the shortcut, we get:
=
8 / 5
+
7 / 5
−
4 / 2

=
16 + 14 − 20 / 10

=
30 − 20 / 10
=
10 / 10
= 1

14. Find the value of log0.12564.
  A.  -3
  B.  -2
  C.  -5
  D.  -1
     
   
View Answer

Shortcut:
logbyax =
x / y
(logba)
If b = a = n, then
lognynx =
x / y

Here, log0.12564
= log2−326
Using the shortcut, we get:
=
6 / 3
log22 = − 2 [∵ log22 = 1]

15. If 100.3010 = 2, then find the value of log0.125125.
  A.  −
704 / 295
  B.  −
703 / 299
  C.  −
687 / 303
  D.  −
699 / 301
     
   
View Answer

Shortcut:
logbyax =
x / y
(logba)
If b = a = n, then
lognynx =
x / y

Here, log0.125125
= log2−353
Using the shortcut, we get:
=
3 / −3
log25
= − log25

∵ 100.3010 = 2
⇒ log102 = 0.3010
∴ log105 = log10
10 / 2

= log1010 − log102 = 1 − 0.3010 = 0.6990
or, − log25 =
log105 / log102
= −
0.6990 / 0.3010
= −
699 / 301

16. If log8x + log4x + log2x = 11, then the value of x is:
  A.  64
  B.  58
  C.  60
  D.  61
     
   
View Answer

Shortcut:
logbyax =
x / y
(logba)
If b = a = n, then
lognynx =
x / y

Here, log8x + log4x + log2x = 11
or, log23x1 + log22x1 + log2x = 11
or,
1 / 3
log2x +
1 / 2
log2x + log2x = 11
or,
(
1 / 3
+
1 / 2
+ 1
)
log2x = 11
or
11 / 6
log2x = 11
or, log2x =
11 x 6 / 11
= 6
∴ x = 26 = 64

17. If logx = log5 + 2log3 −
1 / 2
log25, find the value of x.
  A.  11
  B.  8
  C.  9
  D.  5
     
   
View Answer

Shortcut:
logabn = nlogab
logx = log5 + 2log3 −
1 / 2
log25
= log5 + log32 − log(25)1/2
= log5 + log9 − log5 = log9
∴ x = 9
Note 1: loga(b)(1/n) =
1 / n
logab
Note 2: loga(b)−n = − nlogab

18. If 55 −x = 2x −5, find the value of x.
  A.  9
  B.  5
  C.  7
  D.  11
     
   
View Answer

Shortcut:
logabn = nlogab
Note 1: loga(b)(1/n) =
1 / n
logab
Note 2: loga(b)−n = − nlogab
Here; 55 −x = 2x − 5
or, 55 − x = 2 − (5 − x)
or, (5 − x)log5 = −(5 − x)log2
or, (5 − x)log5 + (5 − x)log2 = 0
or, (5 − x)(log5 + log2) = 0
or, (5 − x)
{
log
10 / 2
+ log2
}
= 0
or, (5 − x){log10 − log2 + log2} = 0
or, 5 − x = 0
∴ x = 5

19. If 2log4x = 1 + log4(x − 1), find the value of x.
  A.  1
  B.  3
  C.  5
  D.  2
     
   
View Answer

Shortcut:
logabn = nlogab
Note 1: loga(b)(1/n) =
1 / n
logab
Note 2: loga(b)−n = − nlogab
Here; 2log4x2 = log44 + log4(x − 1)
or, x2 = 4(x − 1)
or x2 − 4x + 4 = 0
or, (x − 2)2 = 0
∴ x = 2

20. If log3 = 0.477 and (1000)x = 3, then x equals to:
  A.  0.169
  B.  1.149
  C.  0.159
  D.  0.129
     
   
View Answer

Shortcut:
logabn = nlogab
Note 1: loga(b)(1/n) =
1 / n
logab
Note 2: loga(b)−n = − nlogab
Here; (1000)x = 3
or, xlog103 = log3
or, 3x = log3
or x =
log3 / 3
=
0.477 / 3
= 0.159

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