Conditions on the Parameters of Logarithm Function

Q. All the pairs $(x,y)$ that satisfy the inequality $2^{\sqrt{{\sin }^{2}x-2\sin x+5}}\cdot \frac{1}{4^{{\sin }^{2}y}}\le 1$ also satisfy the equation :

1. $2\sin x=\sin y$
2. $\sin x=2\sin y$
3. $\sin x=\left|\sin y\right|$
4. $2\left|\sin x\right|=3\sin y$

Q. For $a>0$ and $a\ne 1,$ the roots of the equation ${\log }_{ax}a+{\log }_x{a}^2+{\log }_{a^{2}x}{a}^3=0$ are

1. $a^{-4/3}$
2. $a^{-3/4}$
3. $a$
4. $a^{-1/2}$

Q. If the equation $2^x+4^y=2^y+4^x$ is solved for $y$ in terms of $x$, where $x<0$, then the sum of the solutions is

1. $x{\log }_2(1-2^x)$
2. $x+{\log }_2(1-2^x)$
3. ${\log }_2(1-2^x)$
4. $x+{\log }_2(1+2^x)$

Q. The complete set of values of $x$ which satisfies the inequation ${\log }_{\left(1.5\right)}\left(\frac{2x-8}{x-2}\right)>0$ is

1. $x\in (2,6)$
2. $x\in (-\infty ,2)\cup (6,\infty )$
3. $x\in (6,\infty )$
4. $x\in (-\infty ,2)\cup (4,\infty )$

Q. Number of integral solutions of the equation
${\log }_{0.5x}\left(x^2\right)-14{\log }_{16x}\left(x^3\right)+40{\log }_{4x}\sqrt{x}=0$ is

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

Q. If ${\log }_3(2x+6)$ is defined, then the complete set of values of $x$ is

1. $(-3,\infty )$
2. $(6,\infty )$
3. $(3,\infty )$
4. $(-6,\infty )$

Q. $\text{For the given inequality find the value of}x?$

Q. If ${\log }_{x-7}(x-4)$ is defined, then

1. $x\in (8,\infty )$
2. $x\in (4,7)$
3. $x\in (4,\infty )-\left\{8\right\}$
4. $x\in (7,\infty )-\left\{8\right\}$

Q. The number of real values of the parameter k for which $(lo{g}_{16}x{)}^2-lo{g}_{16}x+lo{g}_{16}k=0$ with real coefficients will have exactly one solution is

1. 2
2. 1
3. 4
4. None of these

Q.

Solution set of the inequality $lo{g}_7\frac{x-2}{x-3}<0$ is

1. $(-\infty ,2)$

2. $(2,\infty )$

3. $(-\infty ,3)$

4. $(3,\infty )$

Q. The real values of $x$ satisfying ${\log }_{0.5}\left(\frac{x+1}{x+2}\right)\le 1$

1. $(-\infty ,-2)\cup [0,\infty )$
2. $(-\infty ,-2]\cup [0,\infty )$
3. $(-\infty ,-2)\cup (-1,\infty )$
4. $R-[-2,-1]$

Q. For $y=lo{g}_{a}x$ to be defined 'a' must be

1. Any positive real number
2. Any number
3. $\ge e$
4. Any positive real number $\ne 1$

Q. The number of real values of $x,$ satisfying $3^{2{\log }_{3}x}-2x-3=0$ is

1. $0$
2. $1$
3. $2$
4. more than $2$

Q. The domain of the function $\sqrt{\left(lo{g}_{0.5}x\right)}$ is

1. $(1,\infty )$
2. $(0,\infty )$
3. $(0,1]$
4. $(0.5,1)$