Logarithmic Inequalities

Q.

Let $f:R\to R$ be a continuous function such that $f\left(x\right)+f(x+1)=2$ for all $x\in R.$

If $I_1={\int }_0^{8}f\left(x\right)dxand{I}_2={\int }_{-1}^{3}f\left(x\right)dx$, then the value of $I_1+2I_2$ is equal to:

Q.

Maximum value of $x{\left(1-x\right)}^2$ when $0\le x\le 2$, is

1. $\frac{2}{27}$

2. $\frac{4}{27}$

3. $5$

4. $0$

Q. If $lo{g}_{\frac{1}{\sqrt{2}}}(x-1)>2$ then x lies in the interval

1. $(-\infty ,\frac{3}{2})$
2. $(\frac{3}{2},\infty )$
3. $(\frac{3}{2},1)$
4. None of these

Q.

If $f(a+b+1-x)=f\left(x\right)\forall x$, where a and b are fixed positive real numbers, then

$\frac{1}{a+b}{\int }_a^{b}x\left(f\right(x)+f(x+1\left)\right)dx$ is equal to.

1. ${\int }_{a-1}^{b-1}f\left(x\right)dx$

2. ${\int }_{a+1}^{b+1}f(x+1)dx$

3. ${\int }_{a-1}^{b-1}f(x+1)dx$

4. ${\int }_{a+1}^{b+1}f\left(x\right)dx$

Q.

An experiment consists of casting a pair of dice and observing the number that falls uppermost on each die.

Determine the event that the number falling uppermost on one die is a $4$ and the number falling uppermost on the other die is greater than $4$. (Select all that apply.)

$$\begin{gathered} (1,1)(1,2)(1,3)(1,4)(1,5)(1,6) \\ (2,1)(2,2)(2,3)(2,4)(2,5)(2,6) \\ (3,1)(3,2)(3,3)(3,4)(3,5)(3,6) \\ (4,1)(4,2)(4,3)(4,4)(4,5)(4,6) \\ (5,1)(5,2)(5,3)(5,4)(5,5)(5,6) \\ (6,1)(6,2)(6,3)(6,4)(6,5)(6,6) \end{gathered}$$

Q. If $x=lo{g}_{3}5,y=lo{g}_{17}25$, which one of the following is correct?

1. x>y
2. None of these
3. x<y
4. x=y

Q. The set of real values of x satisfying $lo{g}_{\frac{1}{2}}(x^2-6x+12)\ge -2$ is

1. $(-\infty ,2]$
2. $[4,+\infty ]$
3. $Noneofthese$
4. $[2,4]$

Q. If A is the set of all $xϵR$ such that $x^{(logx{)}^2-3logx+1}>1000$, and $A=(a,\infty )$ then $\sqrt{10a}$ will be ___. ( (Base of logx is 10).

Q.

If $\frac{1}{lo{g}_3\pi }+\frac{1}{lo{g}_4\pi }>x$, then the biggest possible integer $x$ can take -

1. 2

2. 4

3. 3

4. 5

Q. $x^{lo{g}_{9x}}>9$ implies -

1. $x\epsilon (0,\infty )$
2. $x\epsilon (0,\frac{1}{9})\cup (9,\infty )$
3. $x>1$
4. $x<2$

Q.

If $lo{g}_{0.5}(x-1)<lo{g}_{0.25}(x-1),$ then x lies in the interval.

1. $(2,\infty )$

2. $(3,\infty )$

3. $(-\infty ,0)$

4. (0, 3)

Q. If $\frac{1}{lo{g}_3\pi }+\frac{1}{lo{g}_4\pi }>x$, then x is equal to

1. 3
2. 4
3. 5
4. 2

Q.

If $x<0$, $y<0$, then $log\left(xy\right)$ is equal to _______________ .

1. log(-x) + log(-y)

2. - log x - log y

3. log x + log y

4. log x - log y

Q.

Solve for $x$:

$lo{g}_3\left(\frac{x}{3}\right)+lo{g}_{1/9}\left(x\right)<1$

1. $(-\infty ,1)$

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

3. None of these.

4. (0, 81)

Q. If A is the set of all $xϵR$ such that $x^{(logx{)}^2-3logx+1}>1000$, and $A=(a,\infty )$ then $\sqrt{10a}$ will be ___. ( (Base of logx is 10).

Q. If $\frac{1}{lo{g}_3\pi }+\frac{1}{lo{g}_4\pi }>x$, then x is equal to

1. 2
2. 3
3. 4
4. 5

Q. Solve the following inequality:
$\sqrt{x^2-3x+2}>2x-5.$

Q.

If A is the set of all $xϵR$ such that $x^{(logx{)}^2-3logx+1}>1000$, and $A=(a,\infty )$ then $\sqrt{10a}$ will be ___. (Base of $logx$ is 10).

Q.

If $x\in R$ satisfies $(lo{g}_{10}100x{)}^2+(lo{g}_{10}10x{)}^2+lo{g}_{10}x\le 14$ then $x$ contains the interval

1. $[{10}^{\frac{-9}{2}},10]$
   

2. $(-1,\infty )$

3. (0, 10)

4. $(0,\infty )$
   

Q. If $l=\log \frac{5}{7},m=\log \frac{7}{9}$ and $n=2(\log 3-\log \sqrt{5})$, then the value of $7^{l+m+n}$ will be

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

Q.

Find $x$ if ${\log }_2(x-5)>3$

1. x<13

2. x>5

3. x>13

4. x>8

Q. $x^{lo{g}_{9}x}>9$ implies -

1. $x\epsilon (0,\frac{1}{9})\cup (9,\infty )$
2. $x>1$
3. $x\epsilon (0,\infty )$
4. $x<2$

Q.

If $x=lo{g}_5\left(1000\right)$ and $y=lo{g}_7\left(2058\right)$ then

1. x > y

2. x < y

3. x = 3 + $lo{g}_{5}8$

4. None of these

Q.

$lo{g}_{3}5\times lo{g}_{25}27=$

1. 3 log 5

2. 2/3

3. 3/2

4. -3/2

Q. Solve the following inequality:
$x>\sqrt{24-5x}.$

Q. Which of the following when simplified, reduces to unity?

1. ${\log }_{10}5.{\log }_{10}20+({\log }_{10}2{)}^2$
2. $\frac{2\log 2+\log 3}{\log 48-\log 4}$
3. $-{\log }_5{\log }_3{\sqrt{9}}^{1/5}$
4. $\frac{1}{6}{\log }_{\sqrt{\frac{3}{4}}}\left(\frac{64}{27}\right)$

Q. If $\frac{1}{lo{g}_3\pi }+\frac{1}{lo{g}_4\pi }>x$, then x be

1. $2$
2. $3$
3. $3.5$
4. $\pi$

Q. If the position vectors of the vertices $A,B$ and $C$ of a $△ABC$ are respectively $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$, then the position vector of the point, where the bisector of $\angle A$ meets $BC$ is :

1. $\frac{1}{2}(4\hat{i}+8\hat{j}+11\hat{k})$
2. $\frac{1}{3}(6\hat{i}+13\hat{j}+18\hat{k})$
3. $\frac{1}{4}(8\hat{i}+14\hat{j}+9\hat{k})$
4. $\frac{1}{3}(6\hat{i}+11\hat{j}+15\hat{k})$

Q. Express each of the following in the form of $\frac{p}{q}$
(i) 0.555.......

Q. $x^{lo{g}_{9}x}>9$ implies -

1. $x\epsilon (0,\infty )$
2. $x\epsilon (0,\frac{1}{9})\cup (9,\infty )$
3. $x>1$
4. $x<2$