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Question Bank No: 1

1. If $a^{x} = bc, b^{y} = ca, c^{z} = ab, then$ $\frac{x}{1 + x} + \frac{y}{1 + y} + \frac{z}{1 + z}$ =

a)		1
b)		2
c)		3
d)		4

2. If 1, $\log_{9} (3^{1 - x} + 2), \log_{3} (4 \cdot 3^{x} - 1)$ are in A.P., then x=

a)		$\log_{3} 4$
b)		${1 - \log}_{3} 4$
c)		$\log_{4} 3$
d)		${1 - \log}_{4} 3$

3. The number of solutions of the equation $\log_{4} (x - 1) = \log_{2} (x - 3)$ is

a)		0
b)		1
c)		2
d)		3

4. The number of values of x such that $\log_{3} 2, \log_{3} (2^{x} - 5), \log_{3} (2^{x} - \frac{7}{2})$ are in A.P. is

a)		0
b)		1
c)		2
d)		3

5. The number $\log_{2} 7$ is

a)		an integer
b)		prime
c)		rational
d)		irrational

6. The number of rational roots of the equation $x^{\frac{3}{4} {(\log_{2} x)}^{2} + \log_{2} x - \frac{5}{4}} = \sqrt{2}$ is

a)		0
b)		1
c)		2
d)		3

7. The number of solutions of the equation $\log_{(2 x + 3)} (6 x^{2} + 23 x + 21) + \log_{(3 x + 7)} (4 x^{2} + 12 x + 9) = 4$ is

a)		0
b)		1
c)		2
d)		>2

8. The number of solutions of $\log_{7} \log_{5} (\sqrt{x + 5} + \sqrt{x}) = 0$ is

a)		0
b)		1
c)		2
d)		>2

9. If $\log_{0.3} (x - 1) < \log_{0.09} (x - 1),$ then $x \in$

a)		$(1, 2)$
b)		$(2, \infty)$
c)		$(- 2, 0)$
d)		none of these

10. For $0 < a < x,$ the minimum value of $\log_{a} x$ + $\log_{x} a$ is

a)		0
b)		1
c)		2
d)		3

11. If n is a natural number such that $n = {p_{1}}^{a_{1}} \cdot {p_{2}}^{a_{2}} \cdot {p_{3}}^{a_{3}} ..... {p_{k}}^{a_{k}}$ , where $p_{1}$ , $p_{2}$ , ...... $p_{k}$ are distinct primes, then log n $\geq$

a)		k
b)		2k
c)		$k^{2}$
d)		k log 2

12. If $x > 1,$ the least value of 2 $\log_{10} x - \log_{x} 0.01$ is

a)		10
b)		2
c)		0.1
d)		4

13. If $a > 0., 2 \log_{x} a + \log_{ax} a + 3 \log_{a^{2} x} a = 0,$ then x=

a)		$a^{\frac{1}{2}}$
b)		$a^{- \frac{1}{2}}$
c)		$a^{- \frac{2}{3}}$
d)		$a^{\frac{4}{3}}$

14. If $4^{x} - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2 x - 1}$ then x=

a)		0
b)		$\frac{1}{2}$
c)		1
d)		$\frac{3}{2}$

15. If $4^{\log_{9} 3} + 9^{\log_{2} 4} = 10^{\log_{x} 83}$ , then x=

a)		2
b)		3
c)		5
d)		10

16. If a, b, c are distinct positive numbers different from 1 such that $(\log_{b} a \log_{c} a - \log_{a} a) + (\log_{c} b \cdot \log_{a} b - \log_{b} b) + (\log_{a} c \log_{b} c - \log_{c} c) = 0,$ then abc=

a)		0
b)		1
c)		2
d)		33

17. Given $\log_{10} 343 = 2.5353,$ the least integer n such that $7^{n} > 10^{10}$ is

a)		10
b)		11
c)		12
d)		13

18. $\sum_{r = 2}^{43} \frac{1}{\log_{r} n}$ =

a)		$\log_{n} 43$
b)		$\log_{43} n$
c)		$\log_{43!} n$
d)		$\frac{1}{\log_{43!} n}$

19. The number of solutions of log2 (x+5) = 6 – x is

a)		2
b)		1
c)		3
d)		None of these

20. The number of log2 7 is

a)		an integer
b)		a rational number
c)		an irrational number
d)		a prime number

21. If 2 log (x + 1) – log ( $x^{2}$ - 1) = log2, then x equals,

a)		1
b)		0
c)		2
d)		3