Logarithmic Inequalities

Q. The domain of the function $f\left(x\right)=\sqrt{({\log }_{0.2}x{)}^3+({\log }_{0.2}{x}^3\left)\right({\log }_{0.2}0.0016x)+36}$ is

1. $[0,625]$
2. $(0,125]$
3. $[0,125]$
4. $(0,625)$

Q. If $x$ satisfies the inequality ${\log }_{x+3}(x^2-x)<1,$ then

1. $x\in (-3,-2)$
2. $x\in (-1,3]$
3. $x\in (1,3)$
4. $x\in (-1,0)$

Q. The solution set of the inequality $\frac{(3^x-4^x)\cdot \ln (x+2)}{x^2-3x-4}\le 0$ is

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

Q. If $3^x=4^{x-1}$, then $x=$

1. $\frac{2{\log }_{3}2}{2{\log }_{3}2-1}$
2. $\frac{2}{2-{\log }_{2}3}$
3. $\frac{1}{1-{\log }_{4}3}$
4. $\frac{2{\log }_{2}3}{2{\log }_{2}3-1}$

Q.

The $n^{th}$ derivative of $x{e}^x$ vanishes when

1. $x=0$

2. $x=-1$

3. $x=-n$

4. $x=n$

Q. Sum of all integral values of $x$ satisfying the inequality
$3^{\frac{5}{2}{\log }_3(12-3x)}-3^{{\log }_{2}x}>32$ is

Q. The domain of $f\left(x\right)={\log }_5(x-[x\left]\right)$ is
(where [.] denotes the greatest integer function)

1. $R-Z$
2. $R$
3. $R^{+}-W$
4. $(0,\infty )$

Q. The value of ${\log }_{5+2\sqrt{6}}(5-2\sqrt{6})$ is

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

Q. The solution set of ${\log }_2|4-5x|>2$ is

1. $(\frac{8}{5},\infty )$
2. $(\frac{4}{5},\frac{8}{5})$
3. $(-\infty ,0)\cup (\frac{8}{5},\infty )$
4. none of these

Q. If $x=lo{g}_{5}3+lo{g}_{7}5+lo{g}_{9}7$, then x $\ge$

1. $\frac{3}{2}$
2. $\frac{1}{2^{\frac{1}{3}}}$
3. $\frac{3}{2^{\frac{1}{3}}}$
4. None

Q.

Evaluate integral $\int e^{3a\log x}+e^{3x\log a}dx$

Q. The correct statement(s) among the following is/are

1. The value of ${\log }_3\left({\log }_{4}65\right)$ is greater than $1$
2. The value of ${\log }_{0.4}0.3$ is less than $1$
3. The value of ${\log }_{0.999}1.999$ is positive
4. The value of ${\log }_{0.6}0.98$ is negative

Q. In a right-angled triangle, $a$ and $b$ are the lengths of the two sides and $c$ is the length of the hypotenuse. If $c+b$ and $c-b$ are numbers other than $1$, then ${\log }_{c+b}a+{\log }_{c-b}a=$

1. ${\log }_{c-b}a\cdot {\log }_{c+b}a$
2. $2{\log }_{c-b}a\cdot {\log }_{c+b}a$
3. $3{\log }_{c-b}a\cdot {\log }_{c+b}a$
4. ${\log }_{(c^2-b^2)}a$

Q. Which of the following numbers has the positive value?

1. ${\log }_{4}12$
2. ${\log }_{0.3}0.1$
3. ${\log }_{3}1.3$
4. ${\log }_{7}0.6$

Q. If ${\log }_{x-2}6+{\log }_{x+2}6>{\log }_{x-2}6\cdot {\log }_{x+2}6$, then $x\in$

1. $(-\infty ,3)$
2. $(-1,3)$
3. $(\sqrt{10},\infty )$
4. $(2,3)$

Q. If ${\log }_{x-3}(x^2-10x+24)\ge {\log }_{x-3}(x^2-6),$ then

1. $x\in (\sqrt{6},4)$
2. $x\in (\sqrt{6},3)$
3. $x\in (\sqrt{6},6)$
4. $x\in (3,4)$

Q. The solution set of ${\log }_3(x^2-2)<{\log }_3(\frac{3}{2}|x|-1)$ contains

1. $(-2,-\sqrt{2})$
2. $(-2,\sqrt{2})$
3. $(\sqrt{2},2)$
4. $(-\sqrt{2},2)$

Q. The complete solution set of the inequality
${\log }_{1/3}(x+1)>{\log }_3(3-x)$ is

1. $x\in (1-\sqrt{3},3)$
2. $x\in (-1,1-\sqrt{3})\cup (1+\sqrt{3},3)$
3. $x\in (-1,1+\sqrt{3})$
4. $x\in (-1,1)\cup (1+\sqrt{3},3)$

Q. The integral value of $x$ satisfying the equation $|{\log }_{\sqrt{3}}x-2|-|{\log }_{3}x-2|=2,$ is

Q. The solution set of the inequality ${\log }_{0.5}\sqrt{\frac{x-4}{x-3}}<{\log }_{0.5}2$ is

1. $(0,3)$
2. $(0,\frac{8}{3})$
3. $(\frac{8}{3},3)$
4. $(\frac{8}{3},3)\cup (4,\infty )$

Q.

If 2|3x+9|<36, then the number of integral values of x is

Q. The domain of the function $f\left(x\right)=2^{{\log }_{2}x}+\frac{2x+3}{x^2-4x+3}$ is

1. $R-\left\{3\right\}$
2. $(0,\infty )-\{1,3\}$
3. $R-\{1,3\}$
4. $R-\left\{1\right\}$

Q. The solution set of the inequality $(0.5{)}^{{\log }_3{\log }_{0.2}(x^2-\frac{4}{5})}<1$ is

1. $(-\frac{2}{\sqrt{5}},\frac{2}{\sqrt{5}})$
2. $(-1,-\frac{2}{\sqrt{5}})\cup (\frac{2}{\sqrt{5}},\infty )$
3. $(-\infty ,-\frac{2}{\sqrt{5}})\cup (\frac{2}{\sqrt{5}},1)$
4. $(-1,-\frac{2}{\sqrt{5}})\cup (\frac{2}{\sqrt{5}},1)$

Q. The complete set of values of $x$ that satisfies the expression $\frac{1}{{\log }_4\left(\frac{x+1}{x+2}\right)}>\frac{1}{{\log }_4(x+3)}$ is

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

Q. The solution set of the inequality ${\log }_3\left(\right(x+2\left)\right(x+4\left)\right)+{\log }_{1/3}(x+2)<\frac{1}{2}{\log }_{\sqrt{3}}7$ is

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

Q. $f:A\to B$ is a function satisfying $3^{f\left(x\right)}+2^{-x}=4$. Then the domain and range of $f$ are

1. $A=\{x\in R:-1<x<\infty \}$;
   $R\left(f\right)=\{x\in R:2<x<4\}$
2. $A=\{x\in R:-3<x<\infty \}$;
   $R\left(f\right)=\{x\in R:0<x<{\log }_{3}4\}$
3. $A=\{x\in R:-2<x<\infty \}$;
   $R\left(f\right)=\{x\in R:-\infty <x<{\log }_{3}4\}$
4. $A=\{x\in R:-2<x<\infty \}$;
   $R\left(f\right)=\{x\in R:-\infty <x<{\log }_{3}41\}$

Q. Let $f\left(x\right)={\int }_0^x(3u^2+2u+2)du,$ where x satisfies the inequality $lo{g}_2(1+\sqrt{6x-x^2-5})⩾0.$ If minimum and maximum values of f(x) are 'm' and 'M' respectively then $\frac{M+m}{4}$ is

Q. Solution set of the inequality ${\log }_3(x+2)(x+4)+{\log }_{\frac{1}{3}}(x+2)<\frac{1}{2}{\log }_{\sqrt{3}}7$ is -

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

Q. Solve $\frac{|x|-1}{|x|-2}\ge 0,x\ne \pm 2$

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

Q. If $x,y,z$ are in G.P., where $x,y,z>1$, then $\frac{1}{1+\log x},\frac{1}{1+\log y},\frac{1}{1+\log z}$ are in

1. A.P.
2. G.P.
3. H.P.
4. None of these