Fundamental Laws of Logarithms

Q.

One Mega-Byte is equal to:

1. 1024 Bytes

2. 1024 Kilo Bytes

3. 1024 Giga Bits

4. 1024 Bits

Q.

If $y={\log }_2{\log }_2\left(x\right)$, then $\frac{dy}{dx}$ is equal to

1. $\frac{{\log }_{2}e}{{\log }_{e}x}$

2. $\frac{{\log }_{2}e}{x{\log }_{x}2}$

3. $\frac{{\log }_{2}x}{{\log }_{e}2}$

4. $\frac{{\log }_2\mathrm{e}}{{\mathrm{xlog}}_{\mathrm{e}}\mathrm{x}}$

Q. The value of $lo{g}_8$ 128 is equal to

1. $\frac{7}{3}$
2. 16
3. $\frac{3}{7}$
4. None

Q. $lo{g}_{5\sqrt{5}}$ 5 is equal to

1. $\frac{2}{3}$
2. $\frac{1}{3}$
3. $\frac{1}{2}$
4. 2

Q. The value of $lo{g}_{b}a.lo{g}_{c}b.lo{g}_a$ c is

1. \N
2. log abc
3. 1
4. 10

Q.

If $x={\log }_{b}a$, $y={\log }_{c}b,$$z={\log }_{a}c$, then $xyz$ is

1. $0$

2. $1$

3. $2$

4. $3$

Q. If ${\log }_{a}3=2;{\log }_{b}8=3$, then ${\log }_{a}b=$

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

Q.

If $a,b,c$ are distinct positive numbers, each different from 1, such that $[{\log }_{b}a{\log }_{c}a-{\log }_{a}a]+[{\log }_{a}b{\log }_{c}b-{\log }_{b}b]+[{\log }_{a}c{\log }_{b}c-{\log }_{c}c]=0,thenabc=$

1. $1$

2. $2$

3. $3$

4. $4$

Q. If $3+{\log }_{5}x=2{\log }_{25}y$, then the value of $x$ is

1. $\frac{y}{25}$
2. $\frac{2y}{125}$
3. $\frac{y}{75}$
4. $\frac{y}{125}$

Q.

Subtract $-{\mathrm{m}}^2+5\mathrm{mn}$ from $4{\mathrm{m}}^2-3\mathrm{mn}+8$.

Q.

If $lo{g}_{10}x=a$, find the value of ${10}^{a-1}$ in terms of x.

Q. 3. find x if log base 2 log base 1/2 log base 3 x>0

Q.

Let one root of $a{x}^2+bx+c=0$ where $a,b,c$ are integers be $3+\sqrt{5}$, then the other root is

1. 3

2. 

3. None of these

4. 

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.

If $f\left(x\right)={\text{log}}_e(1-x)$ and $g\left(x\right)=\left[x\right]$ then find:

(i) $(f+g)\left(x\right)$

(ii) $\left(fg\right)\left(x\right)$

(iii) $\left(\frac{f}{g}\right)\left(x\right)$

(iv) $\left(\frac{g}{f}\right)\left(x\right)$

Also find $(f+g)(-1),\left(fg\right)\left(0\right),\left(\frac{f}{g}\right)(-1),\left(\frac{g}{f}\left(\frac{1}{2}\right)\right)$

Q.

Position of the point (1, 1) with respect to the circle $x^2+y^2-x+y-1=0$ is

1. Upon the circle

2. Inside the circle

3. Outside the circle

4. can't be told

Q. ${\log }_{n}1+{\log }_n(1+\frac{1}{2})+{\log }_n(1+\frac{1}{3})+\dots \dots +{\log }_n(1+\frac{1}{n-1})=$

1. $1-{\log }_{n}2$
2. $1+{\log }_{n}2$
3. ${\log }_2(n-1)$
4. ${\log }_n(n+1)$

Q. $lo{g}_{10}tan{1}^{\circ }+lo{g}_{10}tan{2}^{\circ }+\dots lo{g}_{10}tan{89}^{\circ }$

1. \N
2. 1
3. 2
4. 3

Q. If $\tan {20}^{\circ }=p,then\frac{\tan {160}^{\circ }-\tan {110}^{\circ }}{1+\tan {160}^{\circ }\tan {110}^{\circ }}=$

1. 
2. 
3. 
4. 

Q. $7log\left(\frac{16}{15}\right)+5log\left(\frac{25}{24}\right)+3log\left(\frac{81}{80}\right)$ is equal to

1. \N
2. 1
3. log 2
4. log 3

Q.

Fill in the blank to make the statement true:

${\left(\frac{13}{14}\right)}^5÷{\left(\_\_\_\right)}^2={\left(\frac{13}{14}\right)}^3$

Q.

If x = 1 +${\log }_a$ bc, y = 1 + ${\log }_b$ ca, z = 1 + ${\log }_c$ ab. Find the value of xy + yz + zx.

1. + +

2. + +

3. xyz

4. 1

Q.

If $lo{g}_{3}y=x$ and $lo{g}_{2}z=x$, find ${72}^x$ in terms of y and z

Q. $Thedomainof\frac{\sqrt{2+x}+\sqrt{2-x}}{x}is$

1. [-2, 2]
2. 
3. (-2, 2)
4. R-{0}

Q. ${\sum }_{n=1}^n\frac{1}{lo{g}_{2^n}\left(a\right)}=$

1. $\frac{n(n+1)}{2}lo{g}_{a}2$
2. $\frac{n(n+1)}{2}lo{g}_{2}a$
3. $\frac{(n+1{)}^2{n}^2}{4}lo{g}_{2}a$
4. $Noneofthese$

Q.

The value of the expression $a^2+3a-5$ when $a=2$ is

1. $5$

2. $15$

3. $-5$

Q.

Find the quotient of the identity function by the reciprocal function.

Q. If $A=\{x:x^{{\log }_{\sqrt{x}}2x}=4\}$, then $n\left(A\right)=$

1. $0$
2. $1$
3. $2$
4. infinitely many

Q. The number of values of $x$ satisfying $1+{\log }_5(x^2-9)={\log }_5(x^2+4x+3)$ is

1. $0$
2. $1$
3. $2$
4. infinitely many

Q.

$lo{g}_{b}a=?$

1. $\frac{lo{g}_{10}a}{lo{g}_{10}b}$

2. $\frac{lo{g}_{e}a}{lo{g}_{e}b}$

3. $\frac{lo{g}_{10}b}{lo{g}_{10}a}$

4. $\frac{lo{g}_{e}a}{lo{g}_{e}b}$