Logarithmic Inequalities

Q.

Fill in the blanks to make the statement true:$(-4)+[15+(-3\left)\right]=[-4+15]+\_\_\_\_\_\_$.

Q.

The function t which maps temperature in degree Celsius to temperature in degree Fahrenheit is defined by $t\left(C\right)=\frac{9C}{5}+32.$ Find

(i) t(0) (ii) t(28) (iii) t(-10)

(iv) The value of C when t(C) = 212.

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

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

Q.

The value of $0.0\overline{37}$, where $0.0\overline{37}$ stands for the number $0.0373737\dots$ is

1. $\frac{37}{1000}$

2. $\frac{37}{990}$

3. $\frac{1}{37}$

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

Q.

If $\left\{\begin{array}{ll}\frac{1}{\left|x\right|}& x\ge 1\\ a{x}^2+b& \left|x\right|<1\end{array}\right.$

is differentiable at every point of the domain, then the values of $a$ and $b$ are respectively:

1. $\left(\frac{5}{2},-\frac{3}{2}\right)$

2. $\left(-\frac{1}{2},\frac{3}{2}\right)$

3. $\left(\frac{1}{2},\frac{1}{2}\right)$

4. $\left(\frac{1}{2},-\frac{3}{2}\right)$

Q.

Solution set of ${\log }_3(x^2-2)<{\log }_3(\frac{3}{2}|x|-1)$ is

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

Q.

Prove $1+2+3+\cdots +n<\frac{1}{8}(2n+1{)}^2.$

Q.

In drilling world's deepest hole it was found that the temperature T in degree celcius, x km below the earth's surface was given by T = 30 + 2.5 (x -3), 3 $\le x\le 15$. It was assumed that this drilling resulted

$5\%$ increase in carbon dioxide level at a temperature above

${205}^{\circ }$ and below ${155}^{\circ }C$, so drillers tried to control the depth between these two limits. At what depth will the temperature be between ${155}^{\circ }$ and ${205}^{\circ }$ What values are depicted here by the drillers for selecting this temperature range ?

Q. If ${\log }_{0.04}(x-1)\ge {\log }_{0.2}(x-1),$ then

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

Q. The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c, respectively. Of these subjects, the students has a 75% chance of passing in atleast one, a 50% chance of passing in atleast two and a 40% chance of passing in exactly two. Which of the following relations are true?

1. $p+m+c=\frac{19}{20}$
2. $p+m+c=\frac{27}{20}$
3. $pmc=\frac{1}{10}$
4. $pmc=\frac{1}{4}$

Q. The least value of $2lo{g}_{100}a-lo{g}_a\left(0.0001\right),a>1$ is

1. None
2. 2
3. 4
4. 3

Q. Let f be a function satisfying $2f\left(x\right)-3f\left(\frac{1}{x}\right)=x^2$ for any $x\ne 0$, then the value of f(2) is

1. -2
2. 
3. 4
4. 

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

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

Q.

Which is the larger fraction?

$\frac{17}{102}or\frac{12}{102}$

Q. If ${\log }_{1/2}(4-x)\ge {\log }_{1/2}2-{\log }_{1/2}(x-1),$ then $x\in$

1. $(1,2]$
2. $[3,4)$
3. $(1,3]$
4. $(1,4)$

Q.

Solution set of the inequality $lo{g}_{0.8}\left(lo{g}_6\frac{x^2+x}{x+4}\right)<0$

1. (-4, -3)

2. $(-3,4)\cup (8,\infty )$
   

3. $(-4,-3)\cup (8,\infty )$

4. $(-3,\infty )$
   

Q.

If $a_1$, $a_2$, $a_3$, ........$a_{24}$ are in arithmetic progression and $a_1$ + $a_5$ + $a_{10}$ + $a_{15}$ + $a_{20}$ + $a_{24}$ = 225, then find the value of

$a_1$ + $a_2$ + $a_3$ + ..........+ $a_{23}$ + $a_{24}$

Q. If $lo{g}_{03}(x-1)<lo{g}_{009}(x-1)$, then x lies in the interval

1. $(-2,-1)$
2. $(1,2)$
3. None
4. $(2,\infty )$

Q. If $a,b$ are the maximum and minimum integral values of $x$ which satisfy the inequality $\frac{{\log }_{4}x-2}{{\log }_{4}x-5}<0$ respectively, then the value of $a-b$ is

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. (0, 10)

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

3. $(0,\infty )$
   

4. $(-1,\infty )$

Q.

The set of real values x satisfying ${\log }_{0.3}$ (x-3)>3.

1. x>3

2. x>3.027

3. x > 0.027

4. x ∈ (3, 3.027)

Q. Match the inequalities with their solution sets

1. 11, ∞
2. 1, 109
3. 1, 10
4. (199, ∞)

Q.

Find the set of real values of x for which ${\log }_{0.5}{\log }_2\frac{2x+1}{x+1}>0$

1. x ∈ (-1, )

2. x ∈ (1, )

3. x ∈ (2, )

4. x ∈ (0, )

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=y
4. None

Q.

The number is as much greater than $27$ as it is less than $73$. Find the number.

Q. The solution set of ${\log }_{1/2}{\log }_3\left(\frac{x+1}{x-1}\right)\ge 0$ is

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

Q. The values of $x$ satisfying the inequality $4\left(9^x\right)-19\left(3^x\right)+12\le 0,$ lie in the interval $[A,B].$ Then

1. $A={\log }_3\frac{3}{4}$
2. $A={\log }_{1/3}\frac{4}{3}$
3. $B={\log }_{3}4$
4. $B={\log }_{4}3$

Q. $lo{g}_{(x-2)}(2x-3)>lo{g}_{(x-2)}(24-6x)$
The solution set of the above inequality has integral values of x ___

Q.

If $y=xlog\frac{x}{(a+bx)}$ , then $x^n\frac{d^{2}y}{d{x}^2}={(x\frac{dy}{dx}-y)}^m$, where:

1. 

2. 

3. 

4. 

Q. Three students A, B, C can solve the problem independently, the probabilities of solving the problem of each of them are $\frac{1}{2},\frac{1}{3},\frac{1}{5}$ respectively, then the probability that at least one of them solves the problem is

1. $\frac{1}{30}$
2. $\frac{4}{15}$
3. $\frac{29}{30}$
4. $\frac{11}{15}$