TM

TARGET : JEE (Main + Advanced) 2016JEE (Main

NURTURE COURSE
+ Advanced) 2016
Path to success KOTA (RAJASTHAN)
NURTURE COURSE

SPECIAL RACE # 01 (LOGARITHM) MATH EM ATI CS

Solve the following equations :

1
1. logx–13 = 2 2. log 4 (2log 3 (1 + log 2 (1 + 3log 3 x))) =
2

3. log3(1 + log3(2x – 7)) = 1 4. log3(3x – 8) = 2 – x

5. log 2 (9 - 2x ) 6. log5–x(x2 – 2x + 65) = 2

=1
3- x

æ 1 ö
7. log 3 ç log9 x + + 9x ÷ = 2x
è 2 ø
8. log105 + log10(x +10) –1 = log10(21x–20) – log10(2x–1)

1 æ 1ö æ 1ö 1 æ 1ö
9. log10 x - log10 ç x - ÷ = log10 ç x + ÷ - log10 ç x + ÷
2 è 2ø è 2ø 2 è 8ø
2 5
10. (x - 2)log10 ( x -2) +log10 (x - 2) -12
= 10 2log10 (x -2)
log10 x + 7
1+ log10 x
11. x = 10x 12. x 4
= 10log10 x +1
2
log10 x + log10 x 2 - 2
æ log10 x ö
13. ç ÷ = log10 x 14. 3 log 2 x - log 2 8x + 1 = 0
è 2 ø

15. 2(log x 5) 2 - 3log x 5 + 1 = 0 16. (a log b x )2 - 5x logb a + 6 = 0

æ1ö
17.
2
log10 (100x) + log10
2
(10x) = 14 + log10 ç ÷
èxø

18. log 4 (x 2 - 1) - log 4 (x - 1) 2 = log 4 (4 - x) 2

19. 3 + 2logx +13 = 2log3(x + 1) 20. logx(9x2). log 32 x = 4

ANSWER KEY
1. {1 + 3} 2. {3} 3. {4} 4. {2} 5. {0}
6. {–5} 7. {1/3} 8. {3/2, 10} 9. {1} 10. {2 + 10–7, 3, 102}
11. {10–1, 10} 12. {10–4, 10} 13. {10–3, 10, 100} 14. {2, 16} 15. { 5, 5}
16. {2loga b , 3loga b }, a > 0, a ¹ 1, b > 0, b ¹ 1 17. { 10-9 , 10} 18. {3 + 6}
19. {-(3 - 3) / 3,8} 20. {1/9, 3}
Maths / SPECIAL RACE ON LOGARITHM 1/4
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JEE (Main + Advanced) 2016
Path to success KOTA (RAJASTHAN) NURTURE COURSE

SPECIAL RACE # 02 (LOGARITHM) MATH EM ATI CS

Solve the following equations :
æ x2 ö
1 log1/2 2 (4x) + log 2 ç ÷ = 8 2. log 0.5x x 2 - 14 log16x x 3 + 40log 4x x = 0
è 8 ø
4- 2log 3
3. 6 - (1 + 4.9 3
).log 7 x = log x 7, x Î Q 4. log3(4.3x – 1) = 2x + 1
æ 2+ x ö æ 2 ö
5. log 5 ç ÷ = log 5 ç ÷ 6. 1 + 2log(x+2)5 = log5(x + 2)
è 10 ø è x +1 ø
1 - 2(log10 x 2 ) 2
7. log42 = 24x log 2 4
8. =1
log10 x - 2(log10 x) 2
9. log2(4.3x – 6) – log2(9x – 6) = 1 10. log10(log10x) + log10(log10x4 – 3) = 0
4
11. 2 log 8 (2x) + log 8 (x 2 + 1 - 2x) = 12. log 2 (2x 2 ).log 4 (16x) = log 4 x 3
3
13. log 3 6 - log 3 2 = (log10 x - 2) log 3 12
2 2 2
14. log62x+3 – log6(3x – 2) = x
æ 1 ö
15. ç 1 + ÷ log10 3 + log10 2 = log10 (27 - 3)
x

è 2x ø
SPECIAL RACE # 03 (LOGARITHM) MATH EM ATI CS
Solve the following equations :
1
1. log10 x + 3log10 2 + x = log10 x(x + 2) + 2 2. log 2 (4 x + 1) = x + log 2 (2x +3 - 6)
2
3. log3(9x + 9) = x + log3(28 – 2.3x) 4. log10(log10x) + log10(log10x3 – 2) = 0

5. log 2 (4x + 4) = log 2 2 x + log 2 (2 x +1 - 3)

1 x log10 4 3
3log10
2
x - log10 x
6. log10 (3x - 24 - x ) = 2 + log10 16 - 7. x 3
= 100 3 10
4 2
8. log2(25x+3 – 1) = 2 + log2(5x+3 + 1) 9. log 3 130 - 7 log x (6 - x) = 2
1 9
10. log 2 (4 x +1 + 4).log 2 (4 x + 1) = log1/ 2 11. log 2 (2x 2 ).log 2 (16x) = log 22 x
8 2

12.
æ
log10 4 + ç 1 +
è 2x ø
1 ö
÷ log10 3 = log10 ( x
3 + 27 ) 13. 5log10 x = 50 - x log10 5

2 2
14. | x - 1|log10 x -log10 x = | x - 1|3 15. |x – 10|log2(x – 3) = 2(x – 10)
RACE # 02 ANSWER KEY
1. {2 , 2}
–7
2. {1/ 2,1, 4} 3. {7} 4. {–1, 0} 5. {3}
6. {–9/5, 23} 7. {2} 8. {1/ 10, 3 10} 9. {1} 10. {10}
11. {2} 12. {16} 13. {10- 3 , 10 3 } 14. {log34} 15. xÎÆ
RACE # 03 ANSWER KEY
1. {98} 2. {0} 3. {–1,2} 4. {10} 5. {2} 6. {3}
7. {10-1, 10} 8. {–2} 9. {2} 10. {0} 11. {2–2/5, 16}
12. xÎf 13. {100} 14. {10–1, 2, 103} 15. {13/4, 10}
Maths / SPECIAL RACE ON LOGARITHM 2/3
 TM
JEE (Main + Advanced) 2016
Path to success KOTA (RAJASTHAN) NURTURE COURSE

SPECIAL RACE # 04 (LOGARITHM) MATH EM ATI CS

[SINGAL CORRECT CHOICE TYPE]

1. If log 2 ( 4 + log 3 ( x ) ) = 3 , then sum of digits of x is-
(A) 3 (B) 6 (C) 9 (D) 18
2. Sum of all the solution(s) of the equation log10(x) + log10(x + 2) – log10(5x + 4) = 0 is-
(A) –1 (B) 3 (C) 4 (D) 5
a b
3. If 2 = 3 and 9 = 4 then value of (ab) is-
(A) 1 (B) 2 (C) 3 (D) 4
4. The product of all the solutions of the equation x1+ log10 x = 100000x is-
(A) 10 (B) 105 (C) 10–5 (D) 1

5. If x = log 2 ( )
56 + 56 + 56 + 56 + .......¥ , then which of the following statements holds good ?

(A) x < 0 (B) 0 < x < 2 (C) 2 < x < 4 (D) 3 < x < 4
2
6. If n Î N such that characteristic of n to the base 8 is 2, then number of possible values of n is-
(A) 14 (B) 15 (C) 448 (D) infinite

If x1 & x2 are the two values of x satisfying the equation 7 2x - 2 ( 7x ) + 72x +24 = 0 , then (x1 + x2)
2 2
+ x +12
7.
equals-
(A) 0 (B) 1 (C) –1 (D) 7
8. Number of integral values of x which do not satisfy the equation |x – 1| + |x – 3| = 2|x – 2| is -
(A) 0 (B) 1 (C) 2 (D) more than 2

9. The number of solution(s) of log 3 ( 3x 2 ) .log 9 (81x) = log9 x 3 is-

(A) 0 (B) 1 (C) 2 (D) 3

10. The greatest value of (4log10x – logx(.0001)) for 0 < x < 1 is-
(A) 4 (B) –4 (C) 8 (D) –8
11. The number of integral solutions of | log5 x 2 - 4 |= 2 + | log 5 x - 3 | is-
(A) 1 (B) 2 (C) 3 (D) 0
12. If (x1,y1) and (x2,y2) are solutions of system of equations log289x + logay = 4 and logx289 – logya = 1 such
that log51(x1x2y1y2) = 12 then 'a' equals to-
(A) 729 (B) 243 (C) 81 (D) 27

13. If log a (1 - 1 + x ) = log a 2 ( 3 - 1 + x ) , then number of solutions of the equation is-

(A) 0 (B) 1 (C) 2 (D) infinitely many
n
14. If x, y Î 2 when n Î I and 1 + logxy = log2y, then the value of (x + y) is
(A) 2 (B) 4 (C) 6 (D) 8
3/2 2lnx 4
15. If x1 and x2 are the roots of equation e . x = x , then the product of the roots of the equation is -
2
(A) e (B) e (C) e3/2 (D) e–2

Maths / SPECIAL RACE ON LOGARITHM 3/3

 TARGET : JEE (Main + Advanced) 2016 NURTURE COURSE

[MULTIPLE CORRECT CHOICE TYPE]

16. The equation log x 5 = log 2 5 = a has -

3 x 2 - 6x +11

(A) 3 real solutions for x (B) 4 real solutions for x

29 log 3 5
(C) sum of all real solutions for x is (D) maximum value of 'a' is log 11 - 2
3 3

17. Which of the following statements is(are) correct ?

(A) 71/7 > (42)1/14 > 1 (B) log3(5) log7(9) log11(13) > – 2

1 1
(C) 99 + 70 2 + 99 - 70 2 is rational (D) log 3 + log 3 > 3
4 7

If ( log b a ) + ( log a b ) = 79 , (a > 0, b > 0, a ¹ 1, b ¹ 1 ) then value of (logb a) + (logab) can be-
2 2
18.

(A) 7 (B) –9 (C) 9 (D) –7

19. In which of the following cases the real number 'm' is greater than the real number 'n' ?
(A) m = log345, n = log3004 (B) m = log3004, n = log4003
(C) m = log203, n = log4003 (D) m = log4928, n = log72
æ1ö
log1/ 2 ç ÷ æ 4 ö æ 1 ö
20. The expression 2 è3ø
+ log 2 ç ÷ + log 1 ç ÷ is equals to-
è 11 + 7 ø 2 è 18 + 2 77 ø

æ1ö
log1 / p ç ÷
(A) 7 (B) 7 7 7.......¥ (C) 6 (D) p è 7ø

[SUBJECTIVE]

2 æ a 4 b3 ö
21. ÷ = ap + bp + gp + d (" p Î R – {0}), then
3 2
Given log3a = p = logbc and logb9 = 2 . If log 9 ç
p è c ø
(a+b+g+d) equals
22. If log2(x2 + 1) + log13(x2 + 1) = log2(x2 + 1) log13(x2 + 1), (x ¹ 0) then log7(x2 + 24) is equal to

ANSWER KEY
1. C 2. C 3. A 4. D 5. C 6. B 7. B 8. B 9. B
10. D 11. A 12. A 13. A 14. D 15. A 16. A,D 17. A,B,D
18. B,C 19. A,B,C,D 20. A,B,D 21. 3 22. 2
4/4 Maths / SPECIAL RACE ON LOGARITHM