Logarithms

Q. The solution of the inequality ${\log }_x(2x-\frac{3}{4})>2$

1. $x\in (\frac{3}{8},1)\cup (1,\frac{3}{2})$
2. $x\in (1,\frac{3}{2})$
3. $x\in (\frac{3}{8},\frac{1}{2})$
4. $x\in (\frac{3}{8},\frac{1}{2})\cup (1,\frac{3}{2})$

Q. If the sum of two numbers $p$ and $q$ is $\sqrt{10}$ and their difference is $\sqrt{6}$, then the value of ${\log }_{q}p$ is

1. $-1$
2. $0$
3. $1$
4. $2$

Q. The solution set of the inequality ${\log }_{10}(x^2-16)\le {\log }_{10}(4x-11)$ is

1. $(4,\infty )$
2. $(\frac{11}{4},5)$
3. $(\frac{11}{4},\infty )$
4. $(4,5]$

Q.

$7\log \left(\frac{16}{15}\right)+5\log \left(\frac{25}{24}\right)+3\log \left(\frac{81}{80}\right)$ is equal to

1. $0$

2. $1$

3. $\log 2$

4. $\log 3$

Q. The value of ${\log }_{(\sqrt{3+2\sqrt{2}}+\sqrt{3-2\sqrt{2}})}{2}^9$ is

Q. The set of all values of $x$ satisfying $x^{{\log }_x(1-x{)}^2}=9$ is

1. A subset of $R$ containing $N$
2. A subset of $R$ containing $Z$ (set of all integers).
3. Is a finite set containing at least $2$ elements.
4. A finite set.

Q. The number $lo{g}_{2}7$ is

1. An integer
2. A rational number
3. An irrational number
4. A prime number

Q. If $|a^x-2|+|8-a^x|<5$ , then $x\in$____ , where $a\in (1,\infty )$

1. $\varphi \left(\text{empty set}\right)$
2. $R$
3. $(1,\infty )$
4. $[1,\infty )$

Q. Number of integer(s) in the domain of the function $f\left(x\right)={\cos }^{-1}(x^2-4)$ is

Q. If $f\left(x\right)=x^2$ and $g\left(x\right)=2^x$, then the solution set of the equation $f\left(2^x\right)=g\left(x^2\right)$ is

1. $[0,2]$
2. $R$
3. $\{0,2\}$
4. $R-[0,2]$

Q. Solution set of the inequality $(x-2{)}^{x^2-6x+8}>1$, where $x>2$ is

1. $(2,3)\cup (4,\infty )$
2. $(2,4)$
3. $(3,4)\cup (4,\infty )$
4. $(2,\infty )$

Q. The set of all $x$ satisfying $4^{x^2+2}-9\times 2^{x^2+2}+8=0$ consists of

1. Infinitely many points,
2. Finitely many points from the set of all natural numbers,
3. Finitely many points from the set of all integers
4. Two integers

Q. The number of integral solution(s) of ${\cos }^{-1}(4x^3-12x^2+11x-\frac{5}{2})=\frac{\pi }{3}$ is

1. $0$
2. $3$
3. $1$
4. $2$

Q. The number $lo{g}_{2}7$ is

1. An integer
2. A rational number
3. An irrational number
4. A prime number

Q. If $\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}$, then which of the following is/are true?

1. $xyz=1$
2. $x^a{y}^b{z}^c=1$
3. $x^{b+c}{y}^{c+a}{z}^{a+b}=1$
4. $xyz=x^a{y}^b{z}^c$

Q.

Solution set of the inequality
$\frac{1}{2^x-1}>\frac{1}{1-2^{x-1}}is$

1. $(1,\infty )$

2. $(0,lo{g}_2(4/3\left)\right)$

3. $(-1,\infty )$

4. $(0,lo{g}_2(4/3\left)\right)\cup (1,\infty )$

Q.

If $x{y}^2=4$ and $lo{g}_3\left(lo{g}_{2}x\right)+lo{g}_{1/3}\left(lo{g}_{1/2}y\right)=1$, then x equals

1. 4
   

2. 8
   

3. 18
   
   

4. 64

Q.

If $\frac{lo{g}_2(4x^2-x-1)}{lo{g}_2(x^2+1)}>1$ then x lies in the interval

1. $(-\infty ,-2/3)$
2. $(1,\infty )$
3. $(-2/3,0)$
4. None of these