Logarithms

Q.

Prove that $7\surd 5$ Is irrational number.

Q. <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Two natural numbers differ by 3 . Find the numbers, if the sum of their reciprocals is $7/10$.

Q. The logarithmic form of $100=5^x$ is .

1. $lo{g}_{5}100=x$
2. $lo{g}_{x}100=5$
3. $lo{g}_{100}x=5$

Q.

$5^4=625$ can also be expressed as

1. $lo{g}_{4}5=625$

2. $lo{g}_{5}625=4$

3. $lo{g}_{4}625=5$

4. $\sqrt[4]{4625}=5$

Q.

If $lo{g}_5(3x-2)=2$, then $x$ is equal to

1. 3

2. 15

3. 5

4. 9

Q.

The relation between x, y and z where x = 81, y = 4 and z = 3 written in the exponential form is :

1. z raised to the power of y=x

2. x raised to the power of y=z

3. y raised to the power ofz=x

4. x raised to the power ofz=y

Q.

If $A=\left\{x,y\right\}$, then the power set of $A$ is

1. $\left\{x^x,y^y\right\}$

2. $\left\{ɸ,x,y\right\}$

3. $\left\{ɸ,\left\{x\right\},\left\{2y\right\}\right\}$

4. $\left\{ɸ,\left\{x\right\},\left\{y\right\},\left\{x,y\right\}\right\}$

Q. $\frac{d}{dx}\{e^{x^e}+x^{e^x}+e^{x^x}\}$ =

1. $e^{x^e}+x^{e^x}+e^{x^x}$
   
2. $e{e}^{x^e}{x}^{e-1}+x^{e^x}{e}^x(\frac{1}{x}+logx)+e^{x^x}{x}^x(1+logx)$
3. None
4. $x^2{e}^{x^e}+e^x{x}^{e^x}+x^x{e}^{x^x}$

Q.

$\text{What are the conditions under which the}$
$\text{logarithm}$ $lo{g}_{2x}(x-1)$ is defined ?

1. $x>0$

2. $x>1$

3. $x<-1$

4. $x>1$ and $x<-1$

Q. The solution set of $x^2<3x+10$ is given by the inequality

1. $x>-2$
2. $-2<x<5$
3. $x<5$
4. -2 $\le$ x $\le$ 5
5. $x>5$

Q. Find the logarithms of
$125$ to base $5\surd 5$, and ${}^{.}25$ to base $4$.

Q. If ${\log }_e\left(\frac{a+b}{2}\right)=\frac{1}{2}\left({\log }_{e}a+{\log }_{e}b\right)$ then relation between $a$ and $b$ will be,

Q. What are the conditions under which the logarithm log ${}_{2x}(x^2$ - 1) is defined ?

1. (a) x > 0
2. (b) x > 1
3. (c) x > 1 and x < -1
4. (d) x < -1

Q.

The set of all values of x satisfying $x^{lo{g}_x(1-x{)}^2}=9$ is

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

Q. Logarithmic form of $3^3=27$ is ______?

1. $27={\log }_3^3$
2. $3={\log }_3^{27}$
3. $3={\log }_{27}^3$
4. $27={\log }_{10}^3$

Q. If ${\log }_{1/\sqrt{2}}\frac{1}{\sqrt{8}}={\log }_2(4^x+1)\times {\log }_2(4^{x+1}+4)$, then the value of $x$ will be

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

Q. Solve the following inequality:
$\frac{x^2-5x+6}{|x|+7}<0$

Q. Solve the following inequalities
$x^2-|x|-2⩾0.$

Q. Solve the following inequalities
$f^{\prime }\left(x\right)>g^{\prime }\left(x\right)iff\left(x\right)=x+ln(x-5),g\left(x\right)=ln(x-1)$

Q.

The set of real values of x satisfying the equation
${|x-1|}^{lo{g}_3\left(x^2\right)-2lo{g}_x\left(9\right)}=(x-1{)}^7$
is

1. {2}
   

2. {27, 81}
   

3. {2, 81}

4. {81}
   

Q. if p2 is an even integer , prove that p is an even integer

Q.

What is the exponential form of the logarithmic equation$4={\log }_{0.8}0.4096$?

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. The logarithm of $0.125$ to the base $2$ is equal to

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