Logarithms 2011 Worksheet
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MATHEMATICS : LOGARITHMS : WORKSHEET-I
Multiple Choice Questions with one or more than one correct option 1.
Which of the following when simplified reduces to an integer ? (A)
2.
2 log 6 log12 + log 3
Let N =
log5 16 − log5 4 (C) log5 128
log 32 (B) log 4
1
+ log log abc bc
1 ca
(B) a prime number (D) an integer
+ log abc
(A) 1/2
1 ab
abc
has the value equal to:
(B) 1 1
4.
−2
log3 135 log3 5 − . Then N is : log15 3 log405 3
(A) a natural number (C) a rational number 3.
1 (D) log1/4 16
log5 9 Let, N = 81 + 3 409
(A) 0
3 log
6
3
.
( 7)
(C) 2 2 log25 7
(D) 4
− 125log 25 6 then log N has the value = 2
(B) 1
(C) − 1
(D) none
5.
The number of solution of the equation, log(− 2x) = 2 log (x + 1) is : (A) zero (B) 1 (C) 2 (D) none
6.
If a log b x − 5 x logb a + 6 = 0, where a > 0, b > 0 & ab≠1, then the value of x equals:
(
)
2
(A) 2logb a 7.
(B) 3loga b
The expression, logp logp
(D) a logb 3
(C) b loga 2
p p p
...... p p where p ≥ 2, p ∈ N, when simplified is : 14 4244 3 n radical sign
(A) independent of p, but dependent on n (B) independent of n, but dependent of p (C) dependent on both p & n (D) negative. 8.
If 5 x log2 3 + 3log 2 x = 162 then logarithm of x to the base 4 has the value equal to (A) 2 (B) 1 (C) − 1 (D) 3/2
9.
If a = logx yz, b = logy zx & c = logz xy where x, y, z are positive reals not equal to unity, then abc − a − b − c is equal to : (A) 2 (B) 1 (C) − 1 (D) zero
10.
If log1/3
3x − 1 is less than unity then x must lie in the interval : x+2
(A) (− ∞,− 2)∪ (58 , ∞) (B) (−2, 58) ASSIGNMENT - LOG
(C) (− ∞,− 2)∪ ( 13 ,
5 8
)
(D) (−2, 13) IIT - 2011
11.
12.
The equation logx2 16 + log2x64 = 3 has : (A) one irrational solution (C) two real solutions
(B) no prime solution (D) one integral solution
Assuming all logarithms to be well defined, the value of 1 1 1 equals : + + logbc2 abc log ca 2 abc logab 2 abc
(A) 3
(B) 2
(C)
1 2
(D)
3 2
13.
If a = logx yz, b = logy zx & c = logz xy where x, y, z are positive reals not equal to unity, then abc − a − b − c is equal to : (A) 2 (B) 1 (C) − 1 (D) zero
14.
If 3 2 log3 x − 2 x − 3 = 0, then the number of values of 'x' satisfying the equation is (A) zero (B) 1 (C) 2 (D) more than 2
15.
= 1 is : Number of real solution(s) of the equation x − 3 (A) exactly four (B) exactly three (C) exactly two (D) exactly one
16.
If log1/5 (2x − 4) < log1/5 (x + 3) then : (A) x < 7 (B) 2 < x < 7
3 x 2 − 10 x + 3
(C) x > 7
(D) none
17.
The value(s) of 'b' for which the equation, 2 log1/25 (bx + 28) = − log5 (12 − 4x − x2) has coincident roots, is/are : (A) b = − 12 (B) b = 4 (C) b = 4 or − 12 (D) b = − 4 or 12
18.
The solution set of the equation, 3 log10 x + 2 log10 (A) {10, 10 2}
19.
(B) {10, 103}
If 0º < x < 90º & cos x = (A) 0
ASSIGNMENT - LOG
3 10
(B) 1
1 = 2 is : x
(C) {10, 104}
(D) {10, 102, 104}
then the value of, log10 sinx + log10cos x + log10 tan x is : (C) − 1
(D) none
IIT - 2011
MATHEMATICS : LOGARITHMS : WORKSHEET-I
ANSWERS 1. A,D
2. A,B,C,D
3. B
4. A
5. B
6. B,C
7. A,D
8. D
9. A
10. A
11. C
12. A
13. A
14. B
15. B
16. C
17. B
18. C
19. C
ASSIGNMENT - LOG
IIT - 2011
MATHEMATICS : LOGARITHMS : WORKSHEET-II
State whether True or False with justification 1.
log4 18 is not a rational number.
2.
State whether True or False. log3x . log4x . log5x = log3x . log4x + log4x . log5x + log5x . log3x has two real solutions.
Fill in the blanks with suitable values 1.
Solution set of the equation, log 102 x + log10x2 = log 102 2 − 1 is ______ .
2.
Solve for x . log4 (log2x) + log2 (log4x) = 2 is ______ .
3.
If log147 = a & log14 5 = b, then log35 28 can be expressed in terms of 'a' and 'b' as______ .
4.
Given, logax = α ; logbx = β ; logcx = γ & logdx = δ (x ≠ 1), then logabcd x has the value equal to ______ .
5.
1 log 2
6.
log 3 log 2 log 3 81 has the value equal to ______ .
7.
The expression (0.05)
5
π 5
sin . log
5 simplifies to ______ .
sin π5
log
20
( 0. 1 )
is a perfect square of the natural number ______
(where
0.1 denotes 0.111111 ..... ∞)
(
)
2+ 8 =
1 . Then the value of 1000 x is equal to _____ . 3
8.
If logx log18
9.
The solution set of the system of equations, log3x + log3y = 2 + log32 and log27(x + y) =
2 is ______ . 3
10.
If log3 log4 log32 (x − 3) vanishes, then x can be equal to ______ or ______ .
11.
The ordered pair (x, y) satisfying simultaneously the system of equations ; logx log3 logx y = 0 and logy 27 = 1 is ______ , ______ . loga a 2 − 1 . log12/ a a 2 − 1
If
13.
Which is greater, 2 log12 145 or 15 .
14.
Solve for x , y :
(
)
loga 2 a − 1 . log3 a 6 a − 1 2
ASSIGNMENT - LOG
2
=
1 then the value of 'a' is equal to ______ . 2
12.
log100 x + y =
1 2 IIT - 2011
log10 y − log10 x = log100 4 . 5 x
15.
Solve the equation, log52 x + log5x = 1 .
16.
> 0 . Solve , log0.2 log6 2 x + 1
17.
Find the solution set of the equation, (log5 x)2 + log5x
x2 − x
5 =1. x
MATHEMATICS : LOGARITHMS : WORKSHEET-II
ANSWERS
1. True
1 1 , 20 5
2. True
2−a
2. x = 16
3. a + b
4.
5. 2
6. 1
7. 9
8. 125
9. (6 , 3) & (3 , 6) ]
10.
13 x = − 10 , y = 20
14. 1 , 5 ,
1.
ASSIGNMENT - LOG
28 or 12 ] 9
1 25
11..a =
5 +1 2
α
−1
1 + β + γ −1 + δ −1 −1
12. 2 log12 145
15. (− ∞ , − 1)
IIT - 2011
Multiple Choice Questions with one or more than one correct option 1.
Which of the following when simplified reduces to an integer ? (A)
2.
2 log 6 log12 + log 3
Let N =
log5 16 − log5 4 (C) log5 128
log 32 (B) log 4
1
+ log log abc bc
1 ca
(B) a prime number (D) an integer
+ log abc
(A) 1/2
1 ab
abc
has the value equal to:
(B) 1 1
4.
−2
log3 135 log3 5 − . Then N is : log15 3 log405 3
(A) a natural number (C) a rational number 3.
1 (D) log1/4 16
log5 9 Let, N = 81 + 3 409
(A) 0
3 log
6
3
.
( 7)
(C) 2 2 log25 7
(D) 4
− 125log 25 6 then log N has the value = 2
(B) 1
(C) − 1
(D) none
5.
The number of solution of the equation, log(− 2x) = 2 log (x + 1) is : (A) zero (B) 1 (C) 2 (D) none
6.
If a log b x − 5 x logb a + 6 = 0, where a > 0, b > 0 & ab≠1, then the value of x equals:
(
)
2
(A) 2logb a 7.
(B) 3loga b
The expression, logp logp
(D) a logb 3
(C) b loga 2
p p p
...... p p where p ≥ 2, p ∈ N, when simplified is : 14 4244 3 n radical sign
(A) independent of p, but dependent on n (B) independent of n, but dependent of p (C) dependent on both p & n (D) negative. 8.
If 5 x log2 3 + 3log 2 x = 162 then logarithm of x to the base 4 has the value equal to (A) 2 (B) 1 (C) − 1 (D) 3/2
9.
If a = logx yz, b = logy zx & c = logz xy where x, y, z are positive reals not equal to unity, then abc − a − b − c is equal to : (A) 2 (B) 1 (C) − 1 (D) zero
10.
If log1/3
3x − 1 is less than unity then x must lie in the interval : x+2
(A) (− ∞,− 2)∪ (58 , ∞) (B) (−2, 58) ASSIGNMENT - LOG
(C) (− ∞,− 2)∪ ( 13 ,
5 8
)
(D) (−2, 13) IIT - 2011
11.
12.
The equation logx2 16 + log2x64 = 3 has : (A) one irrational solution (C) two real solutions
(B) no prime solution (D) one integral solution
Assuming all logarithms to be well defined, the value of 1 1 1 equals : + + logbc2 abc log ca 2 abc logab 2 abc
(A) 3
(B) 2
(C)
1 2
(D)
3 2
13.
If a = logx yz, b = logy zx & c = logz xy where x, y, z are positive reals not equal to unity, then abc − a − b − c is equal to : (A) 2 (B) 1 (C) − 1 (D) zero
14.
If 3 2 log3 x − 2 x − 3 = 0, then the number of values of 'x' satisfying the equation is (A) zero (B) 1 (C) 2 (D) more than 2
15.
= 1 is : Number of real solution(s) of the equation x − 3 (A) exactly four (B) exactly three (C) exactly two (D) exactly one
16.
If log1/5 (2x − 4) < log1/5 (x + 3) then : (A) x < 7 (B) 2 < x < 7
3 x 2 − 10 x + 3
(C) x > 7
(D) none
17.
The value(s) of 'b' for which the equation, 2 log1/25 (bx + 28) = − log5 (12 − 4x − x2) has coincident roots, is/are : (A) b = − 12 (B) b = 4 (C) b = 4 or − 12 (D) b = − 4 or 12
18.
The solution set of the equation, 3 log10 x + 2 log10 (A) {10, 10 2}
19.
(B) {10, 103}
If 0º < x < 90º & cos x = (A) 0
ASSIGNMENT - LOG
3 10
(B) 1
1 = 2 is : x
(C) {10, 104}
(D) {10, 102, 104}
then the value of, log10 sinx + log10cos x + log10 tan x is : (C) − 1
(D) none
IIT - 2011
MATHEMATICS : LOGARITHMS : WORKSHEET-I
ANSWERS 1. A,D
2. A,B,C,D
3. B
4. A
5. B
6. B,C
7. A,D
8. D
9. A
10. A
11. C
12. A
13. A
14. B
15. B
16. C
17. B
18. C
19. C
ASSIGNMENT - LOG
IIT - 2011
MATHEMATICS : LOGARITHMS : WORKSHEET-II
State whether True or False with justification 1.
log4 18 is not a rational number.
2.
State whether True or False. log3x . log4x . log5x = log3x . log4x + log4x . log5x + log5x . log3x has two real solutions.
Fill in the blanks with suitable values 1.
Solution set of the equation, log 102 x + log10x2 = log 102 2 − 1 is ______ .
2.
Solve for x . log4 (log2x) + log2 (log4x) = 2 is ______ .
3.
If log147 = a & log14 5 = b, then log35 28 can be expressed in terms of 'a' and 'b' as______ .
4.
Given, logax = α ; logbx = β ; logcx = γ & logdx = δ (x ≠ 1), then logabcd x has the value equal to ______ .
5.
1 log 2
6.
log 3 log 2 log 3 81 has the value equal to ______ .
7.
The expression (0.05)
5
π 5
sin . log
5 simplifies to ______ .
sin π5
log
20
( 0. 1 )
is a perfect square of the natural number ______
(where
0.1 denotes 0.111111 ..... ∞)
(
)
2+ 8 =
1 . Then the value of 1000 x is equal to _____ . 3
8.
If logx log18
9.
The solution set of the system of equations, log3x + log3y = 2 + log32 and log27(x + y) =
2 is ______ . 3
10.
If log3 log4 log32 (x − 3) vanishes, then x can be equal to ______ or ______ .
11.
The ordered pair (x, y) satisfying simultaneously the system of equations ; logx log3 logx y = 0 and logy 27 = 1 is ______ , ______ . loga a 2 − 1 . log12/ a a 2 − 1
If
13.
Which is greater, 2 log12 145 or 15 .
14.
Solve for x , y :
(
)
loga 2 a − 1 . log3 a 6 a − 1 2
ASSIGNMENT - LOG
2
=
1 then the value of 'a' is equal to ______ . 2
12.
log100 x + y =
1 2 IIT - 2011
log10 y − log10 x = log100 4 . 5 x
15.
Solve the equation, log52 x + log5x = 1 .
16.
> 0 . Solve , log0.2 log6 2 x + 1
17.
Find the solution set of the equation, (log5 x)2 + log5x
x2 − x
5 =1. x
MATHEMATICS : LOGARITHMS : WORKSHEET-II
ANSWERS
1. True
1 1 , 20 5
2. True
2−a
2. x = 16
3. a + b
4.
5. 2
6. 1
7. 9
8. 125
9. (6 , 3) & (3 , 6) ]
10.
13 x = − 10 , y = 20
14. 1 , 5 ,
1.
ASSIGNMENT - LOG
28 or 12 ] 9
1 25
11..a =
5 +1 2
α
−1
1 + β + γ −1 + δ −1 −1
12. 2 log12 145
15. (− ∞ , − 1)
IIT - 2011
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